The automorphisms of class two groups of prime exponent
aa r X i v : . [ m a t h . G R ] J a n The automorphisms of class two groups ofprime exponent
Michael Vaughan-LeeJanuary 2015
Abstract
In 2012, Marcus du Sautoy and Michael Vaughan-Lee gave an ex-ample of a class two group G p of prime exponent p and order p , andthey showed that the number of descendants of G p of order p is nota PORC function of p . The fact that the number of descendants of G p is not PORC is directly related to the fact that the order of theautomorphism group of G p is not PORC. The number of conjugacyclasses of G p is also not a PORC function of p . In this note we givea complete list of all class two groups of prime exponent with order p k for k ≤
8. For every group in this list we are able to show thatthe number of conjugacy classes of the group is polynomial in p , andthat the order of the automorphism group is also polynomial in p .Thus, in some sense, the group G p is minimal subject to having anon-PORC number of conjugacy classes and a non-PORC number ofautomorphisms. Graham Higman wrote two immensely important and influential papers onenumerating p -groups in the late 1950s. The papers were entitled Enumerat-ing p -groups I and II, and were published in the Proceedings of the LondonMathematical Society in 1960 (see [2] and [3]). In the first of these papersHigman proves that if we let f ( p n ) be the number of p -groups of order p n ,then f ( p n ) is bounded by a polynomial in p . In the second of his two papersHigman formulated his famous PORC conjecture concerning the form of thefunction f ( p n ). He conjectured that for each n there is an integer N (de-pending on n ) such that for p in a fixed residue class modulo N the function1 ( p n ) is a polynomial in p . For example, for p ≥ p is3 p + 39 p + 344 + 24 gcd( p − ,
3) + 11 gcd( p − ,
4) + 2 gcd( p − , . (See [4].) So for p ≥ f ( p ) is one of 8 polynomials in p , with the choiceof polynomial depending on the residue class of p modulo 60. Thus f ( p ) is P olynomial O n R esidue C lasses. The various nineteenth century classifica-tions of groups of order p n for n ≤ f ( p n ) is PORC for n ≤ p [6] shows that f ( p ) is PORC. Itis still an open question whether f ( p ) is PORC, but in a recent article [8] Ishowed that the function enumerating the number of groups of order p withexponent p is PORC. However Marcus du Sautoy and I have found a classtwo group G p of order p and exponent p with the property that the numberof class 3 groups H of order p such that H/γ ( H ) ∼ = G p is not PORC.It may still be the case that f ( p ) is PORC, but this example does raisea strong possibility that Higman’s conjecture fails for n = 10. The detailsof this example, and a history of the PORC conjecture can be found in [1].Here we give some of the main properties of G p .The group G p is a six generator class two group of exponent p ( p > G p = (cid:28) x , x , x , x , x , x , y , y , y | [ x , x ] = y , [ x , x ] = y , [ x , x ] = y [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = y (cid:29) where all other commutators are defined to be 1, and where all generatorshave order p . The main results in [1] are Theorem 1
Let D p be the number of descendants of G p of order p andexponent p . Let V p be the number of solutions ( x, y ) in GF ( p ) that satisfy x + 6 x − and y = x − x .1. If p = 5 mod 12 then D p = ( p + 1) / .2. If p = 7 mod 12 then D p = ( p + 1) / .3. If p = 11 mod 12 then D p = ( p + 1) / p + 1) / .4. If p = 1 mod 12 and V p = 0 then D p = ( p + 1) / .5. If p = 1 mod 12 and V p = 0 then D p = ( p − /
36 + ( p − / . heorem 2 There are infinitely many primes p = 1 mod 12 for which V p > . However there is no sub-congruence of p = 1 mod 12 for which V p > forall p in that sub-congruence class. So the number of descendants of G p of order p and exponent p is notPORC, and it easily follows that the number of descendants of order p withno restriction on the exponent is also not PORC.We also prove in [1] that the order of the automorphism group of G p isas follows:1. If p = 5 mod 12 there are | GL(2 , p ) | · p automorphisms.2. If p = 7 mod 12 there are | GL(2 , p ) | · p automorphisms.3. If p = 11 mod 12 there are | GL(2 , p ) | · p automorphisms.4. If p = 1 mod 12 and V p = 0 there are | GL(2 , p ) | · p automorphisms.5. If p = 1 mod 12 and V p = 0 there are | GL(2 , p ) | · p automorphisms.So the order of the automorphism group of G p is not PORC. To see whythe order of the automorphism group of G p impacts on the number of descen-dants of G p we need to briefly recall the p -group generation algorithm [5]. Let P be a p -group. The p -group generation algorithm uses the lower p -centralseries, defined recursively by P ( P ) = P and P i +1 ( P ) = [ P i ( P ) , P ] P i ( P ) p for i ≥
1. The p -class of P is the length of this series. Each p -group P , apartfrom the elementary abelian ones, is an immediate descendant of the quotient P/R where R is the last non-trivial term of the lower p -central series of P .Thus all the groups with order p n , except the elementary abelian one, areimmediate descendants of groups with order p k for k < n . All of the immedi-ate descendants of P are quotients of a certain extension of P (the p -coveringgroup); the isomorphism problem for these descendants is equivalent to theproblem of determining orbits of certain subgroups of this extension underan action of the automorphism group of P .The group G p also has the property that the function enumerating thenumber of its conjugacy classes is not PORC. I showed in [7] that the numberof conjugacy classes of G p is p + p − p − p − p + 1) × E, where E is the number of points on the elliptic curve y = x − x over GF( p )(including the point at infinity). I give a proof in [7] of the well known factthat E is not PORC. 3t would be interesting to find more examples of finite p -groups with anon-PORC number of conjugacy classes, or a non-PORC number of auto-morphisms, and so I undertook a systematic search of all the class two groupsof exponent p and order p n for n ≤
8. My search found nothing of interest!
Theorem 3
For every prime p > there are class two groups of exponent p and order p n with n ≤ . The number of conjugacy classes of each of thesegroups is polynomial in p , and the number of automorphisms of each of thesegroups is polynomial in p . For every prime p > p of order p , one of order p , three of order p , seven of order p , fifteen of order p , and forty-three of order p . We give presentations for all these groupsbelow. For example the single class two group of exponent p and order p has presentation h a, b | class2 , exponent p i . (So it is the free group of rank 2 in the variety of class two groups of exponent p .) The single class two group of exponent p and order p has presentation h a, b, c | [ c, a ] , [ c, b ] , class2 , exponent p i . Note that the prime p in these presentations is a parameter, so that thetwo presentations define families of groups of order p and p — one groupin each family for each prime value of p >
2. However it is easier to think ofthe presentations as defining a single group with the prime p undetermined.The first of these groups has p + p − p + p − p conjugacy classes.Many of the presentations involve a second parameter ω , which is assumedthroughout to be an integer which is primitive modulo p . For example oneof the class two groups of exponent p and order p has presentation h a, b, c, d | [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] = [ b, a ] ω i . (For the remainder of this note we will omit the words “class 2, exponent p ”from the presentations, taking them as understood.) In this presentation wecan take ω to be any integer which is not a quadratic residue modulo p . Theisomorphism type of the group does not depend on the particular choice of4on-quadratic residue. However, for consistency in all the different presen-tations it is convenient to assume throughout that ω is always a primitiveelement modulo p .Finally, two of the presentations involve a parameter m which takes avalue such that x + mx − p ) in one of the presen-tations and takes a value such that x − mx + 1 is irreducible in the otherpresentation. The isomorphism types of the two groups do not depend onthe choice of m .We give presentations for each of the 70 class two groups of exponent p and order p n with n ≤
8, and for each of these groups we give the polynomialsgiving the number of conjugacy classes and the order of the automorphismgroup. We also give a description of the automorphism group which givesthe reader enough information to “write down” a set of generators for theautomorphism group. If G is a class two group of exponent p , and if A is theautomorphism group of G then A induces a group of automorphisms on G/G ′ .If G/G ′ has rank k then the full automorphism group of G/G ′ is GL( k, p ),and so the automorphisms induced by A on G/G ′ form a subgroup, B say, ofGL( k, p ). Furthermore this subgroup B of GL( k, p ) completely determines A . To see this, suppose that the defining generators of G are a , a , . . . , a k and let α ∈ A . If we pick arbitrary g , g , . . . , g k ∈ G ′ then there is anautomorphism mapping a i to ( a i α ) g i for i = 1 , , . . . , k . So B completelydetermines A , and | A | = | B | · | G ′ | k .We give three examples of the induced automorphism groups B . For ourfirst example we consider the group h a, b, c, d | [ c, a ] , [ c, b ] , [ d, a ] , [ d, b ] , [ d, c ] i of order p . The action of the automorphism group on G/G ′ is given bymatrices in GL(4 , p ) of the form ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . This is intended to show that the matrices in B must have zeros in the fourpositions shown, but that their other entries are arbitrary subject to therestriction that the matrix must have non-zero determinant. If we write thematrix in block form (cid:18) X Y Z (cid:19) × X and Z must lie inGL(2 , p ), whereas Y is arbitrary. So the group of all these matrices has order( p − ( p − p ) p . Perhaps the simplest way to generate the group is to take generators with ran-dom entries in all the ∗ positions, checking that the determinant is non-zero,and continue adding generators until they generate a subgroup of GL(4 , p ) ofthe right order. Alternatively, take generators (cid:18) ω
00 1 (cid:19) and (cid:18) − − (cid:19) forGL(2 , p ) (where ω is a primitive element in GF( p )). Then the matrix group B is generated by the following matrices: ω , − − , ω
00 0 0 1 , − − , , , , . As a second example, consider the group h a, b, c, d | [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] = [ b, a ] ω i of order p . The action of the automorphism group on G/G ′ is given bymatrices in GL(4 , p ) of the form α β γ δε ζ η θ − ωθ ωη ζ − εωδ − ωγ − β α , and α β γ δε ζ η θωθ − ωη − ζ ε − ωδ ωγ β − α where ( α, β, γ, δ ) can be any 4-vector other than zero ( p − ε, ζ , η, θ ) can be any 4-vector which is not in the linear span6f ( α, β, γ, δ ) and ( ωδ, − ωγ, − β, α ) ( p − p possibilities). Again, one way ofgenerating this group of matrices is to throw in random non-singular matricesof the form shown as generators until the required order 2( p − p − p ) isreached. Alternatively, take the matrix − − as the first generator, and then throw in random non-singular matrices ofthe first form as generators until the required order is reached. A secondalternative is to first find generators for the subgroup of matrices of the firstkind with first row (1 , , , −
10 0 0 1 , − ω , ζ η ωη ζ
00 0 0 1 with ζ , η not both zero. Furthermore, the matrices (cid:18) ζ ηωη ζ (cid:19) with ζ , η notboth zero form a group of order p − p ). So this group is cyclic, and it is easy to find a single elementwhich generates the group. So we can find three matrices which generate thegroup of matrices of the first type with first row (1 , , , − − and with matrices of the first type with general first row ( α, β, γ, δ ), andsecond row (0 , , ,
0) if α = 0 or δ = 0, or second row (1 , , ,
0) if α = δ = 0.Experimentally it seems that we only need one of these matrices — the onewith first row (0 , , ,
0) and second row (0 , , , h a, b, c, d | [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] i
7f order p . The action of the automorphism group on G/G ′ is given bymatrices in GL(4 , p ) of the form α β ∗ ∗ γ δ ∗ ∗ λδ − λγ − λβ λα with λ ( αδ − βγ ) = 0. Here (cid:18) α βγ δ (cid:19) takes arbitrary values in GL(2 , p ), λ takes any non-zero value, and the entries in the four positions denoted ∗ takearbitrary values. This group of matrices has order ( p − p − p − p ) p ,and it easy to see that if we let ω be a primitive element in GF( p ) then it isgenerated by the matrices ω
00 0 0 ω , ω ω , − − − − , , , , . The complete list of 70 class two groups of exponent p and order p n with n ≤ G/G ′ . The information shows that in somesense the automorphism groups are independent of p , though of course theentries in the matrices must lie in GF( p ). Also, to find sets of generatorsfor the matrix groups we need to make some choice of primitive elements inGF( p ) and GF( p ). There is one case, group 8.5.9, where we need differentgenerators for the matrix group when p = 3, but in all other cases the choiceof generators is independent of p , except in the sense just described above.The proofs of the results below are all traditional “hand proofs”, albeitwith machine assistance with linear algebra over rational function fields ofcharacteristic zero. But all the results have been checked with a computerfor small primes. 8 Order p Group 3.2.1 h a, b i The number of conjugacy classes is p + p −
1, and the automorphismgroup has order ( p − p − p ) p .The action of the automorphism group on G/G ′ is given by GL(2 , p ). p Group 4.3.1 h a, b, c | [ c, a ] , [ c, b ] i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p − p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(3 , p ) of the form ∗ ∗ ∗∗ ∗ ∗ ∗ . p Group 5.3.1 h a, b, c | [ c, b ] i The number of conjugacy classes is 2 p − p , and the automorphism grouphas order ( p − p − p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(3 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ . .4 Four generator groups Group 5.4.1 h a, b, c, d | [ c, a ] , [ c, b ] , [ d, a ] , [ d, b ] , [ d, c ] i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 5.4.2 h a, b i × [ b,a ]=[ d,c ] h c, d i The number of conjugacy classes is p + p −
1, and the automorphismgroup has order ( p − p )( p − p )( p − p ) p .The centre of the group has order p . The image of a can be anythingoutside the centre ( p − p choices). The image of b can be anything outsidethe centralizer of the image of a ( p − p choices). The image of c mustcentralize the images of a and b and lie outside the centre ( p − p choices).The image of d must centralize then images of a and b , but not the image of c , and must be scaled so that the image of [ d, c ] equals the image of [ b, a ] ( p choices). p Group 6.3.1 h a, b, c i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p − p )( p − p ) p .The action of the automorphism group on G/G ′ is given by GL(3 , p ).10 .4 Four generator groups Group 6.4.1 h a, b, c, d | [ c, b ] , [ d, a ] , [ d, b ] , [ d, c ] i The number of conjugacy classes is 2 p − p and the order of the auto-morphism group is ( p − ( p − p − p ) p . The action of the automorphism group on
G/G ′ is given by matrices inGL(4 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 6.4.2 h a, b, c, d | [ c, b ] , [ d, a ] , [ d, b ] = [ b, a ] , [ d, c ] i The number of conjugacy classes is p + 2 p − p − p + 1 and the orderof the automorphism group is 2( p − ( p − p ) p . The action of the automorphism group on
G/G ′ is given by matrices inGL(4 , p ) of the form − α β − γ − α + δε ζ ε η θα γ α with ( αζ − γε )( βθ − δη ) = 0 and α − β γ α − δ ε ζη θ η β δ with ( αθ − γη )( βζ − δε ) = 0. Group 6.4.3 h a, b, c, d | [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] i p + p − p and the order of theautomorphism group is ( p − p − p − p ) p . The action of the automorphism group on
G/G ′ is given by matrices inGL(4 , p ) of the form α β ∗ ∗ γ δ ∗ ∗ λδ − λγ − λβ λα with λ ( αδ − βγ ) = 0. Group 6.4.4 h a, b, c, d | [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] = [ b, a ] ω i The number of conjugacy classes is p + p − p − p − p ) p . The action of the automorphism group on
G/G ′ is given by matrices inGL(4 , p ) of the form α β γ δε ζ η θωθ − ωη − ζ ε − ωδ ωγ β − α and α β γ δε ζ η θ − ωθ ωη ζ − εωδ − ωγ − β α , where ( α, β, γ, δ ) can be any 4-vector other than zero ( p − ε, ζ , η, θ ) can be any 4-vector which is not in the linear span of( α, β, γ, δ ) and ( ωδ, − ωγ, − β, α ) ( p − p possibilities). Group 6.5.1 h a, b i × h c i × h d i × h e i The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − p − p )( p − p − p )( p − p ) p . G/G ′ is given by matrices inGL(5 , p ) of the form ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 6.5.2 h a, b i × [ b,a ]=[ d,c ] h c, d i × h e i The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − p )( p − p )( p − p ) p ( p − p ) . The centre of the group has order p . The image of a can be anythingnot in the centre ( p − p choices). The image of b can be anything whichdoes not centralize the image of a ( p − p choices). The image of c mustcentralize the images of a and b but lie outside the centre ( p − p choices).The image of d must centralize the images of a and b , but not the image of c , and must be scaled so that the image of [ d, c ] equals the image of [ b, a ] ( p choices). The image of e must lie in the centre, but not in the derived group( p − p choices). p Group 7.4.1 h a, b, c, d | [ c, b ] , [ d, b ] , [ d, c ] i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p − p − p )( p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . roup 7.4.2 h a, b, c, d | [ d, a ] , [ d, b ] , [ d, c ] i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p − p − p )( p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ . Group 7.4.3 h a, b, c, d | [ d, a ] , [ c, b ] , [ d, c ] i The number of conjugacy classes is 3 p − p + p , and the automorphismgroup has order 2( p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗
00 0 0 ∗ and ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ . Group 7.4.4 h a, b, c, d | [ d, a ] = [ c, b ] , [ d, b ] , [ d, c ] i The number of conjugacy classes is 2 p + p − p , and the automorphismgroup has order ( p − p − p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form λ ∗ ∗ ∗ α β ∗ γ δ ∗ λ − ( αδ − βγ ) , λ ( αδ − βγ ) = 0. Group 7.4.5 h a, b, c, d | [ c, a ] , [ d, a ] = [ c, b ] , [ d, b ] i The number of conjugacy classes is 2 p + p − p , and the automorphismgroup has order ( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form λα λβ µα − µβ − λγ − λδ − µγ µδνα νβ ξα − ξβνγ νδ ξγ − ξδ , with ( αδ − βγ )( λξ − µν ) = 0 Group 7.4.6 h a, b, c, d | [ d, b ] = [ c, a ] ω , [ d, c ] = [ b, a ] , [ c, b ] i The number of conjugacy classes is p +2 p − p − p , and the automorphismgroup has order 2( p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form α ∗ ∗ β γ ωδ δ γ ωβ ∗ ∗ α and α ∗ ∗ β γ ωδ − δ − γ − ωβ ∗ ∗ − α , with α and β not both zero, and with γ and δ not both zero. Group 7.5.1 h a, b, c, d, e | [ c, b ] , [ d, a ] , [ d, b ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is 2 p − p and the order of the auto-morphism group is ( p − p − ( p − p ) p . G/G ′ is given by matrices inGL(5 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 7.5.2 h a, b, c, d, e | [ c, b ] , [ d, a ] , [ d, b ] = [ b, a ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + 2 p − p − p + p and the orderof the automorphism group is 2( p − p − ( p − p ) p . The action of the automorphism group on
G/G ′ is given by matrices inGL(5 , p ) of the form − α β − γ − α + δ ∗ ε ζ ε ∗ η θ ∗ α γ α ∗ ∗ with ( αζ − γε )( βθ − δη ) = 0 and α − β γ α − δ ∗ ε ζ ∗ η θ η ∗ β δ ∗ ∗ with ( αθ − γη )( βζ − δε ) = 0. Group 7.5.3 h a, b, c, d, e | [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − ( p − p − p ) p . G/G ′ is given by matrices inGL(5 , p ) of the form α β ∗ ∗ ∗ γ δ ∗ ∗ ∗ λδ − λγ ∗ − λβ λα ∗ ∗ . with λ ( αδ − βγ ) = 0. Group 7.5.4 h a, b, c, d, e | [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] = [ b, a ] ω , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + p − p and the order of theautomorphism group is 2( p − p − p − p ) p . The action of the automorphism group on
G/G ′ is given by matrices inGL(5 , p ) of the form α β γ δ ∗ ε ζ η θ ∗ ωθ − ωη − ζ ε ∗− ωδ ωγ β − α ∗ ∗ and α β γ δ ∗ ε ζ η θ ∗− ωθ ωη ζ − ε ∗ ωδ − ωγ − β α ∗ ∗ , where ( α, β, γ, δ ) can be any 4-vector other than zero ( p − ε, ζ , η, θ ) can be any 4-vector which is not in the linear span of( α, β, γ, δ ) and ( ωδ, − ωγ, − β, α ) ( p − p possibilities). Group 7.5.5 h a, b, c, d, e | [ c, b ] , [ d, a ] , [ d, b ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] = [ b, a ] i The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − ( p − p − p ) p .17he action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form α β γ δ ε α − ( λξ − µν ) 0 0 00 ζ η α − ( − δµ + ελ ) 0 λ µ α − ( − δξ + εν ) 0 ν ξ with α, η, λξ − µν = 0. Group 7.5.6 h a, b, c, d, e | [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] = [ b, a ] i The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − p − p − p ) p .The action of the automorphism group on G/G ′ is given by a subgroup H of GL(5 , p ), where the matrices in H have the form α ∗ ∗ β ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ ∗ ∗ δ ∗ ∗ ∗ ∗ . As we run through the elements of H , (cid:18) α βγ δ (cid:19) takes all values in GL(2 , p ).If we take generators (cid:18) ω
00 1 (cid:19) and (cid:18) − − (cid:19) for GL(2 , p ) then we obtainthe following generating matrices for H : ω α β γ ωλ λ ω − γ − ω − α δ ω λ ( λ = 0)and − α β γ λ λ λ − δ β − δ δ − α − λ λ − λ ( λ = 0) . .6 Six generator groups Group 7.6.1 h a, b i × h c i × h d i × h e i × h f i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p )( p − p )( p − p )( p − p )( p − p )( p − p ). Group 7.6.2 h a, b i × [ b,a ]=[ d,c ] h c, d i × h e i × h f i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p )( p − p )( p − p ) p ( p − p )( p − p ). Group 7.6.3 h a, b i × [ b,a ]=[ d,c ]=[ f,e ] h c, d i × [ b,a ]=[ d,c ]=[ f,e ] h e, f i The number of conjugacy classes is p + p −
1, and the automorphismgroup has order ( p − p )( p − p )( p − p ) p ( p − p ) p . p Group 8.4.1 h a, b, c, d | [ b, a ] , [ c, a ] i The number of conjugacy classes is 2 p + p − p , and the automorphismgroup has order ( p − ( p − p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 8.4.2 a, b, c, d | [ b, a ] , [ d, c ] i The number of conjugacy classes is p + 3 p − p − p + p , and theautomorphism group has order 2( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ and ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ . Group 8.4.3 h a, b, c, d | [ b, a ] , [ d, b ][ c, a ] i The number of conjugacy classes is p + 2 p − p − p , and the automor-phism group has order ( p − p − p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form α β γ δ ∗ ∗ − λδ λγ ∗ ∗ λβ − λα , with λ ( αδ − βγ ) = 0. Group 8.4.4 h a, b, c, d | [ d, b ][ c, a ] , [ d, c ][ b, a ] ω i The number of conjugacy classes is p + p − p , and the automorphismgroup has order 2( p − p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(4 , p ) of the form α β γ δε ζ η θ − ωθ ωη ζ − εωδ − ωγ − β α and α β γ δε ζ η θωθ − ωη − ζ ε − ωδ ωγ β − α .
20n these two matrices the first row can be anything non-zero. The first rowdetermines the fourth row up to sign, and the second row can be anythingnot in the span of rows one and four. The subgroup of matrices with firstrow (1 , , ,
0) is generated by matrices with second row ( ε, ζ , η, θ ) with one(or both) of ζ , η non-zero. The full matrix group is then generated by thissubgroup and matrices with a general first row ( α, β, γ, δ ), and second row(0 , , ,
0) if α = 0 or δ = 0, or second row (1 , , ,
0) if α = δ = 0. Group 8.5.1 h a, b, c, d, e | [ e, a ] , [ c, b ] , [ d, b ] , [ e, b ] , [ d, c ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − ( p − p − p )( p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 8.5.2 h a, b, c, d, e | [ d, a ] , [ e, a ] , [ d, b ] , [ e, b ] , [ d, c ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p − p )( p − p − p )( p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 8.5.3 h a, b, c, d, e | [ d, a ] , [ e, a ] , [ c, b ] , [ e, b ] , [ d, c ] , [ e, c ] , [ e, d ] i p − p + p , and the automorphismgroup has order 2( p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ and ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 8.5.4 h a, b, c, d, e | [ d, a ] = [ c, b ] , [ e, a ] , [ d, b ] , [ e, b ] , [ d, c ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is 2 p + p − p , and the automorphismgroup has order ( p − ( p − p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form λ ∗ ∗ ∗ ∗ α β ∗ ∗ γ δ ∗ ∗ λ − ( αδ − βγ ) ∗ ∗ , with λ ( αδ − βγ ) = 0. Group 8.5.5 h a, b, c, d, e | [ c, a ] , [ d, a ] = [ c, b ] , [ e, a ] , [ d, b ] , [ e, b ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is 2 p + p − p , and the automorphismgroup has order ( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form λα λβ µα − µβ ∗− λγ − λδ − µγ µδ ∗ να νβ ξα − ξβ ∗ νγ νδ ξγ − ξδ ∗ ∗ , αδ − βγ )( λξ − µν ) = 0 Group 8.5.6 h a, b, c, d, e | [ d, b ] = [ c, a ] ω , [ d, c ] = [ b, a ] , [ e, a ] , [ c, b ] , [ e, b ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + 2 p − p − p , and the automor-phism group has order 2( p − p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form α ∗ ∗ β ∗ γ ωδ ∗ δ γ ∗ ωβ ∗ ∗ α ∗ ∗ and α ∗ ∗ β ∗ γ ωδ ∗ − δ − γ ∗− ωβ ∗ ∗ − α ∗ ∗ , with α and β not both zero, and with γ and δ not both zero. Group 8.5.7 h a, b, c, d, e | [ e, b ] = [ c, a ][ d, b ] m , [ d, c ] = [ b, a ] , [ e, c ] = [ d, b ] , [ d, a ] , [ e, a ] , [ c, b ] , [ e, d ] i , where m is chosen so that x + mx − p ). Differentchoices of m give isomorphic groups. Note that the discriminant of x + mx − − m −
27 and this must be a square in GF( p ) — we let u = − m − p + p − p , and the automorphismgroup has order 3( p − p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ where the first row can be anything non-zero of the form shown (( p − , , , , , p ) of order 3( p − p generated by matricesof the following form α β γ δ − γ β mα + δ − α β = 0, and u − m m u +92 m − ( u +3) m u u − u +94 u m u − ( u +1) m u ( u +3) m u − u +2 u +94 u . (The cube of the second matrix has the form of the first, with α = γ = δ = 0,and β = 4 u m − .) The full subgroup of GL(5 , p ) giving the action of theautomorphism group on G/G ′ is generated by the matrices above togetherwith p − α β γ γ α − β − mγβ + mγ − γ α − mβ − m γ , where α, β, γ are not all zero. 24 roup 8.5.8 h a, b, c, d, e | [ d, a ][ c, b ] ω , [ e, a ] , [ e, b ] = [ c, a ] , [ d, b ] , [ d, c ] = [ b, a ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + p − p , and the automorphismgroup has order 2( p − p − p .The action of the automorphism group on G/G ′ is given by a group H ofmatrices in GL(5 , p ) of the form ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗ ∗ . The group H has a subgroup of order p ( p −
1) consisting of matrices of theform µξλ ξ − µ ν − λξµ ωλ ξ − − ωλ ξ + µ ξ ξ with ξ = 0.If we premultiply a general matrix ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗ ∗ in H by a suitable matrix from this subgroup of order p ( p −
1) we can obtaina matrix of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ , and the most general matrices of this form arising in the action of the auto-morphism group on G/G ′ are 25 α ωβ α − ωβ β α α − ωβ
00 0 0 0 1 and ξ α ωβ − α + ωβ − β − α α − ωβ
00 0 0 0 1 . There are 2( p − p −
1) of these matrices, and so the order of the subgroupof GL(5 , p ) generated by all these matrices is 2 p ( p − p − Group 8.5.9 h a, b, c, d, e | [ d, a ] , [ e, a ] , [ e, b ] = [ c, a ] , [ d, b ] , [ d, c ] = [ b, a ] , [ e, c ] , [ e, d ] = [ c, b ] i The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p + 1)( p − p .The action of the automorphism group on G/G ′ is given by a subgroup H of GL(5 , p ) of order ( p + 1)( p − p . There is a subgroup K of H of order p ( p − consisting of matrices of the form λ α αβ β α β − α β − αβ α − β
00 0 0 1 0 α β αβ − α β β α with α, λ = 0, and K consists of all the matrices in H with fourth row ascalar multiple of (0 , , , , p = 3 there is a matrix A = H , and when p = 3, there is a matrix A = −
32 227 −
12 19 −
32 227 −
29 32 22714 −
14 118 38 154 in H . In both cases H is generated by A and K . (There are ( p − K or AK .) Group 8.5.10 h a, b, c, d, e | [ d, a ] , [ e, a ] , [ d, b ] , [ e, b ] = [ c, a ] , [ d, c ] = [ b, a ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + 2 p − p − p , and the automor-phism group has order ( p − p − p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form ∗ ∗ ∗∗ α β ∗ ∗∗ γ δ ∗ ∗∗ ∗ ∗∗ ∗ ∗ with αδ − βγ = 0. Rows 2 and 3 are arbitrary subject to the conditionthat αδ − βγ = 0, and rows 2 and 3 then determine rows 1, 4 and 5 upto multiplication by a scalar. (The same scalar for each of the three rows.There is a subgroup of this group of matrices of order ( p − p consisting ofmatrices of the form λ ∗ ∗ ∗∗ ∗ ∗ λ
00 0 0 0 λ ( λ = 0) , and the full subgroup of GL(5 , p ) giving the action of the automorphismgroup on G/G ′ is generated by these matrices, together with the matrices λ λ λ
00 0 0 0 1 ( λ = 0)and − − − − −
10 0 0 − . roup 8.5.11 h a, b, c, d, e | [ d, a ] , [ e, a ] , [ c, b ] , [ e, b ] = [ c, a ] , [ d, c ] = [ b, a ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + 2 p − p − p , and the automor-phism group has order ( p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form λ εη βεη + εζα β γ δ − γε ε − αε − βγεζ η θ ε η with λ, ε, η = 0. Group 8.5.12 h a, b, c, d, e | [ e, a ] , [ c, b ] , [ d, b ] , [ e, c ] , [ e, d ] , [ d, c ] = [ b, a ] , [ e, b ] = [ c, a ] i The number of conjugacy classes is p + 2 p − p − p , and the automor-phism group has order ( p − p .The action of the automorphism group on G/G ′ is given by a group H ofmatrices in GL(5 , p ) where H is generated by matrices of the form λ α β α − β α
00 0 0 0 1 ( α, β, λ = 0)and ∗ γ γ ∗ γ ∗ γ . Group 8.5.13 h a, b, c, d, e | [ d, a ] , [ e, a ] , [ c, b ] , [ e, b ][ d, b ] ω = [ c, a ] , [ d, c ] = [ b, a ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + 2 p − p − p , and the automor-phism group has order 2( p − ( p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form λ α ωβ β − ε − δ γδ ωε εβ α ζ α − ωζ and λ α ωβ βε − δ − γδ ωε ε − β − α − ζ − α + ωζ , α and β are not both zero and ζ = αω − , λ = 0. Group 8.5.14 h a, b, c, d, e | [ c, b ] , [ d, b ] , [ e, c ] , [ e, d ] , [ d, c ] = [ b, a ] , [ e, b ] = [ c, a ] , [ e, a ] = [ d, a ] ω i The number of conjugacy classes is p + 2 p − p − p , and the automor-phism group has order 2( p − p − p .The action of the automorphism group on G/G ′ is given by a subgroup H = KL of GL(5 , p ), where K is a group of matrices of order 2( p − p − λ ( α + ωβ ) 0 0 2 ωλαβ λαβ α β ωβ α λαβ λα λβ ωλαβ ω λβ λα and λ ( α + ωβ ) 0 0 2 ωλαβ λαβ α β − ωβ − α − λαβ − λα − λβ − ωλαβ − ω λβ − λα , with λ = 0 and α, β not both zero, and where L is a normal subgroup of H of order p consisting of matrices ωα β α ωβ . Group 8.5.15 h a, b, c, d, e | [ d, a ] , [ e, a ] , [ c, b ] , [ d, b ] , [ e, c ] , [ d, c ] = [ b, a ] , [ e, b ] = [ c, a ] i The number of conjugacy classes is p + 2 p − p − p , and the automor-phism group has order 2( p − p . 29he action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form λ α β γδ − γ − βδ − δ −
00 0 − αδ δ and λ α β γ γδ − − βδ δ − βδ − δ − , with γ, δ, λ = 0. Group 8.5.16 h a, b, c, d, e | [ d, a ] , [ e, a ] , [ c, b ] , [ d, b ] , [ e, b ] = [ c, a ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + 3 p − p − p + p , and theautomorphism group has order ( p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form α β α − βδ γ
00 0 0 0 δ ε − θ ζ η θ , with α, β, γ, δ = 0. Group 8.5.17 h a, b, c, d, e | [ e, b ] = [ c, a ] , [ d, c ] = [ b, a ][ d, a ] − , [ e, a ] , [ c, b ] , [ d, b ] , [ e, c ] , [ e, d ] i The number of conjugacy classes is p + 3 p − p − p + p , and theautomorphism group has order 2( p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form λ α α α α
00 0 0 0 1 − γ εβ γ β δ γ − ββ β + βγ − γε γ βε − β λ α α α α
00 0 0 0 1 γ ε γ − βγ − γ δ − − ββ + ε β + ε − βγ − γ − βε − β , with α, γ, λ = 0. Group 8.5.18 h a, b, c, d, e | [ e, c ][ e, b ] , [ c, a ] , [ d, a ] , [ e, a ] , [ c, b ] , [ d, b ] , [ e, d ] i The number of conjugacy classes is p + 4 p − p − p + 2 p , and theautomorphism group has order 6( p − p .The action of the automorphism group on G/G ′ is generate by matricesin GL(5 , p ) of the form α β β γ
00 0 0 0 δ ε ζ η − ζ ( α, β, γ, δ = 0) , − and . Here the matrices of the first kind above form a normal subgroup of order( p − p . Group 8.5.19 h a, b, c, d, e | [ d, a ] , [ e, a ] , [ c, b ] , [ d, b ] , [ e, b ] , [ d, c ] , [ e, c ] i The number of conjugacy classes is 2 p + 2 p − p − p + p , and theautomorphism group has order ( p − p − ( p − p ) p .31he action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 8.5.20 h a, b, c, d, e | [ d, c ] = [ b, a ] , [ c, a ] , [ d, a ] , [ e, a ] , [ d, b ] , [ e, b ] , [ e, d ] i The number of conjugacy classes is 2 p + p − p , and the automorphismgroup has order ( p − p .The action of the automorphism group on G/G ′ is given by matrices inGL(5 , p ) of the form α β γ αβγ −
00 0 0 0 δ ε ∗ ∗ ∗∗ ε ∗ ∗ ∗ , with α, β, γ, δ = 0. 32 roup 8.5.21 h a, b, c, d, e | [ d, c ] = [ b, a ] , [ c, a ] , [ d, a ] , [ e, a ] , [ c, b ] , [ d, b ] , [ e, d ] i The number of conjugacy classes is 2 p + p − p , and the automorphismgroup has order ( p − ( p − p − p ) p .The action of the automorphism group on G/G ′ is given by the subgroupof GL(5 , p ) generated by matrices of the form α β γ αβγ −
00 0 0 0 δ ( α, β, γ, δ = 0)and ∗ ε − ε ∗
00 0 0 1 0 ∗ ∗ , − ε − ζ ∗ − ε + ζ − ∗ ∗ (If we let λ, µ, ν, ξ be the (1 , , , ,
4) entries in a general matrixin this group of matrices, then (cid:18) λ µν ξ (cid:19) takes all values in GL(2 , p ).)
Group 8.5.22 h a, b, c, d, e | [ d, c ] = [ b, a ] , [ c, a ] , [ d, a ] , [ e, a ] , [ c, b ] , [ d, b ] , [ e, b ] i The number of conjugacy classes is 2 p + p − p , and the automorphismgroup has order ( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by the subgroupof GL(5 , p ) consisting of matrices of the form ∗ ∗ ∗ ∗ ∗ ∗
00 0 ∗ ∗
00 0 ∗ ∗ ∗ , with the restriction that if we let A be the elements of GL(2 , p ) in posi-tions (1 , , , ,
2) and if we let B be the element of GL(2 , p ) inpositions (3 , , , , A = det B .33 .6 Six generator groups For all these groups we take the generators to be a, b, c, d, e, f , and we justgive the relations, with the class two and exponent p conditions understood. Group 8.6.1 [ c, b ] , [ d, a ] , [ d, b ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ]The number of conjugacy classes is 2 p − p and the order of the auto-morphism group is ( p − p − p − p )( p − p − p )( p − p ) p . The action of the automorphism group on
G/G ′ is given by matrices inGL(6 , p ) of the form ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Group 8.6.2 [ c, b ] , [ d, a ] , [ d, b ] = [ b, a ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ]The number of conjugacy classes is p + 2 p − p − p + p and the orderof the automorphism group is 2( p − ( p − p ) p . The action of the automorphism group on
G/G ′ is given by matrices inGL(6 , p ) of the form − α β − γ − α + δ ∗ ∗ ε ζ ε ∗ ∗ η θ ∗ ∗ α γ α ∗ ∗ ∗ ∗ ∗ ∗ αζ − γε )( βθ − δη ) = 0 and α − β γ α − δ ∗ ∗ ε ζ ∗ ∗ η θ η ∗ ∗ β δ ∗ ∗ ∗ ∗ ∗ ∗ with ( αθ − γη )( βζ − δε ) = 0. Group 8.6.3 [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ]The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − p − ( p − p ) p . The action of the automorphism group on
G/G ′ is given by matrices inGL(6 , p ) of the form α β ∗ ∗ ∗ ∗ γ δ ∗ ∗ ∗ ∗ λδ − λγ ∗ ∗ − λβ λα ∗ ∗ ∗ ∗ ∗ ∗ with λ ( αδ − βγ ) = 0. Group 8.6.4 [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] = [ b, a ] ω , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ]The number of conjugacy classes is p + p − p and the order of theautomorphism group is 2( p − p − p )( p − p − p ) p . G/G ′ is given by matrices inGL(6 , p ) of the form α β γ δ ∗ ∗ ε ζ η θ ∗ ∗ ωθ − ωη − ζ ε ∗ ∗− ωδ ωγ β − α ∗ ∗ ∗ ∗ ∗ ∗ and α β γ δ ∗ ∗ ε ζ η θ ∗ ∗− ωθ ωη ζ − ε ∗ ∗ ωδ − ωγ − β α ∗ ∗ ∗ ∗ ∗ ∗ , where ( α, β, γ, δ ) can be any 4-vector other than zero ( p − ε, ζ , η, θ ) can be any 4-vector which is not in the linear span of( α, β, γ, δ ) and ( ωδ, − ωγ, − β, α ) ( p − p possibilities). Group 8.6.5 [ c, b ] , [ d, a ] , [ d, b ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] = [ b, a ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ]The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − ( p − p − p ) p .The action of the automorphism group on G/G ′ is given by matrices inGL(6 , p ) of the form α β γ δ ε ∗ α − ( λξ − µν ) 0 0 0 ∗ ζ η ∗ α − ( − δµ + ελ ) 0 λ µ ∗ α − ( − δξ + εν ) 0 ν ξ ∗ ∗ with α, η, λξ − µν = 0. 36 roup 8.6.6 [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] = [ b, a ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ]The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − ( p − p − p ) p .The action of the automorphism group on G/G ′ is given by a subgroup H of GL(6 , p ), where the matrices in H have the form α ∗ ∗ β ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ ∗ ∗ δ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . As we run through the elements of H , (cid:18) α βγ δ (cid:19) takes all values in GL(2 , p ).If we take generators (cid:18) ω
00 1 (cid:19) and (cid:18) − − (cid:19) for GL(2 , p ) then we obtainthe following generating matrices for H : ω α β γ ∗ ωλ ∗ λ ∗ ω − γ − ω − α δ ∗ ω λ ∗ ∗ ( λ = 0)and − α β γ ∗ λ λ ∗ λ ∗− δ β − δ δ − α ∗ − λ λ − λ ∗ ∗ ( λ = 0) . roup 8.6.7 [ c, a ] , [ c, b ] , [ d, a ] , [ d, b ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ] = [ b, a ]The number of conjugacy classes is p + p − p + p − p + 1 and theorder of the automorphism group is ( p − p − p )( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by a subgroup H of GL(6 , p ), where the matrices in H have the form ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ . The first row takes on all possible p − g = a ∗ b ∗ e ∗ f ∗ . The centralizerof g has index p in G , and the second row must correspond to an element g = a ∗ b ∗ e ∗ f ∗ outside the centralizer of g ( p − p possibilities). The fifthrow must correspond to a non-trivial element g = a ∗ b ∗ e ∗ f ∗ which centralizes g and g ( p − g = a ∗ b ∗ e ∗ f ∗ which centralizes g and g but does not centralize g ( p − p possibilities), but we require [ g , g ] = [ g , g ] and this reduces the number ofchoices for g to p . The third and fourth rows correspond to non-commutingelements of the form c ∗ d ∗ (( p − p − p ) possibilities). Group 8.6.8 [ c, a ] , [ c, b ] , [ d, a ] , [ d, b ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ] = [ b, a ][ d, c ]The number of conjugacy classes is p + 3 p − p − p + 2 and the orderof the automorphism group is 6( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by a subgroup H of GL(6 , p ). The group H has a subgroup K of order ( p − ( p − p ) p ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , where the determinants of the three 2 × H is generated by K and the matrices and − − . Group 8.6.9 [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ] = [ b, a ]The number of conjugacy classes is p + 2 p − p − p + 1 and the orderof the automorphism group is ( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by the subgroupof GL(6 , p ) consisting of matrices α β ε ζ γ δ η θ λδ − λα − λβ λα ρ σ τ ϕ with αδ − βγ = ρϕ − στ = 0, λ = 0, αη + βθ = γε + δζ .39 roup 8.6.10 [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ] = [ c, a ]The number of conjugacy classes is p + p − p + p − p and the order ofthe automorphism group is ( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by the subgroup H of GL(6 , p ) where the matrices in H have the form ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . There is a subgroup of H of order p consisting of matrices of the form α β γ δ ε ζ η θ δ θ − γ − η . The group H is generated by these matrices together with matrices of theform α β γ δ λδ − λγ − λβ λα ρ σ τ ϕ where ρϕ − στ = λ ( αδ − βγ ) = 0. Group 8.6.11 [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] = [ b, a ] ω , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] , [ f, e ] = [ b, a ]40he number of conjugacy classes is p + p − p and the order of theautomorphism group is 2( p − p − p )( p − p .The action of the automorphism group on G/G ′ is given by the matricesin GL(6 , p ) with the form α β γ δ ε ζ η θ ωθ − ωη − ζ ε − ωδ ωγ β − α λ µ ν ξ and α β γ δ ε ζ η θ − ωθ ωη ζ − ε ωδ − ωγ − β α λ µ ν ξ where αη + βθ = γε + δζ and λξ − µν = αζ − βε + ω ( γθ − δη ) = 0. Notethat there are p − p − p choices for row 2. Group 8.6.12 [ c, b ] , [ d, a ] , [ d, b ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] = [ b, a ] , [ f, a ] , [ f, b ] , [ f, c ] , [ f, d ] = [ c, a ] , [ f, e ]The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − ( p − p ) p .The action of the automorphism group on G/G ′ is given by a subgroup H of GL(6 , p ), where the elements of have the form α ∗ ∗ β ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ ∗ ∗ δ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ where (cid:18) α βγ δ (cid:19) takes all values in GL(2 , p ). There is a subgroup of H oforder p consisting of all elements of the form ε ζ η θ η θ λ µ , H is generated by this subgroup together with elements α β δλ δµ − γλ − γµ δν δξ − γν − γξγ δ − βλ − βµ αλ αµ − βν − βξ αν αξ with ( αδ − βγ )( λξ − µν ) = 0. Group 8.6.13 [ b, a ] , [ d, a ] , [ e, a ][ c, a ] , [ f, a ] , [ c, b ] , [ d, b ] = [ c, a ] , [ e, b ] , [ f, b ][ c, a ] , [ d, c ] , [ e, c ] , [ e, d ] = [ f, c ] , [ f, d ] , [ f, e ] = [ c, a ][ f, c ]The number of conjugacy classes is p + p − p and the order of theautomorphism group is ( p − p − p − p ) p .The action of the automorphism group on G/G ′ is given by a subgroup H = KL of GL(6 , p ), where K is a subgroup of H and L is a normal subgroupof H . The subgroup K is the set of all matrices of the form α β γ δ λ δ − λ γ − λ β λ α λ − λ ) δ (2 λ + λ ) γ λδ λγ λ + 2 λ ) β λ − λ ) α λβ λα with λ ( αδ − βγ ) = 0, and L is the set of all matrices of the form α − β + 2 β + ε − ζ + γδ β + 1 γ β γ − β + 2 β + ε − ζ + γδ δ − η δ β + 1 δ β − α − γ ε − β − γ − β − γζ η − δ − β − δ − β . roup 8.6.14 [ c, b ] , [ d, a ] , [ d, b ] = [ c, a ] , [ d, c ] , [ e, a ] , [ e, b ] , [ e, c ] , [ e, d ] = [ b, a ] , [ f, a ] , [ f, b ] = [ c, a ] m , [ f, c ] = [ b, a ] , [ f, d ] , [ f, e ] = [ c, a ] , where x − mx + 1 is irreducible over GF( p ). (Different choices of m giveisomorphic groups.)The number of conjugacy classes is p + p − p − p − p .Note that since x − mx + 1 is irreducible over GF( p ) its discriminant4 m −
27 must be a square, and we let u = 4 m − G/G ′ is given by a subgroup H GL(6 , p ). The first rows of the matrices in H are completely arbitrary,except that they must be non-zero. The subgroup K of H consisting ofthose matrices in H with first row (1 , , , , ,
0) has order 3( p − p and isgenerated by the following matrices λ λ λ
00 0 0 0 0 1 , − mγ − β − γ − αα β γ − γ mβ − α − β and A = u (9 − u ) m u mu ( u −
3) 00 − mu − u (9 + u ) 0 m u m u (3 − u ) 0 0 u (9 − u ) 0 − m u m u mu − u (9 + u ) . (Here A = I .)The group H is generated by the matrices above, together with ( p − p − p ) matrices of the form B = α β γ δ ε ζµ + mρ λ mξ + σ ρ − ξ ν − ν ξ λ − µ σ − ρζ ε − β α − mδ − γ − mε δρ − σ ξ ν − mµ λ + mσ µδ + mζ − γ − mβ − ε ζ β α . B determines rows 4 and 6, and that the third of B determines rows 2 and 5. The first row can take any non-zero value ( p − B is fixed the entries λ, µ, ν, ξρ, σ from row3 are required to satisfy two homogeneous linear equations with coefficientsdetermined by the first row. These two linear equations correspond to therequirement that the group elements corresponding to rows 3 and 4 commute.There are p solutions to these two equations, but we also require that thethird row lie outside the subspace spanned by rows 1, 4 and 6. Any row3 lying in this subspace automatically satisfies these two linear equations,since the group elements corresponding to rows 1, 4 and 6 are guaranteed tocommute by the choice of rows 4 and 6. So, once row 1 of B is fixed, thereare p − p choices for row 3 of B , and then all the rows of B are determined.It follows from all this that H has order 3( p − p − p .If we want to obtain explicit generators for H then we only need to findone possible third row of B for any given first row. Then these particularchoices of B , together with the generators of K will generate H . If one of α, δ, ζ is non-zero, then we can take the third row of B to be(0 , δ + mζ δ − αζ , − α + mδα + δζ , , ζ − αδ, , and if one of β, γ, ε is non-zero, then we can take the third row of B to be( γ + mεγ − βε, , , γε + β , , ε + mβε + βγ ) . Experimentally, it seems that that H is generated by the generators of K together with the single matrix − − m −
10 0 − m . Group 8.7.1 h a, b i × h c i × h d i × h e i × h f i × h g i . The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p )( p − p )( p − p )( p − p )( p − p )( p − p )( p − p ).44 roup 8.7.2 h a, b i × [ b,a ]=[ d,c ] h c, d i × h e i × h f i × h g i . The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p )( p − p )( p − p ) p ( p − p )( p − p )( p − p ). Group 8.7.3 h a, b i × [ b,a ]=[ d,c ]=[ f,e ] h c, d i × [ b,a ]=[ d,c ]=[ f,e ] h e, f i × h g i . The number of conjugacy classes is p + p − p , and the automorphismgroup has order ( p − p )( p − p )( p − p ) p ( p − p ) p ( p − p ).45 eferences [1] Marcus du Sautoy and Michael Vaughan-Lee, Non-PORC behaviour of aclass of descendant p -groups , J. Algebra (2012), 287–312.[2] G. Higman, Enumerating p -groups. I: Inequalities , Proc. London Math.Soc. (3) 10 (1960), 24–30.[3] G. Higman, Enumerating p -groups. II: Problems whose solution is PORC ,Proc. London Math. Soc. (3) 10 (1960), 566–582.[4] M.F. Newman, E.A. O’Brien, and M.R. Vaughan-Lee, Groups and nilpo-tent Lie rings whose order is the sixth power of a prime , J. Algebra (2004), 383–401.[5] E.A. O’Brien,
The p -group generation algorithm , J. Symbolic Comput. (1990), 677–698.[6] E.A. O’Brien and M.R. Vaughan-Lee, The groups with order p for oddprime p , J. Algebra (2005), 243–358.[7] Michael Vaughan-Lee, Graham Higman’s PORC conjecture , Jahres.Dtsch. Math.-Ver. (2012), 89–106.[8] Michael Vaughan-Lee,
Groups of order p and exponent p , Int. J. GroupTheory4