Rational group algebras of finite groups: from idempotents to units of integral group rings
aa r X i v : . [ m a t h . R A ] J a n RATIONAL GROUP ALGEBRAS OF FINITE GROUPS: FROMIDEMPOTENTS TO UNITS OF INTEGRAL GROUP RINGS
ERIC JESPERS, GABRIELA OLTEANU, AND ´ANGEL DEL R´IO
Abstract.
We give an explicit and character-free construction of a complete set of orthogonalprimitive idempotents of a rational group algebra of a finite nilpotent group and a full descriptionof the Wedderburn decomposition of such algebras. An immediate consequence is a well-knownresult of Roquette on the Schur indices of the simple components of group algebras of finitenilpotent groups. As an application, we obtain that the unit group of the integral group ring Z G of a finite nilpotent group G has a subgroup of finite index that is generated by three nilpotentgroups for which we have an explicit description of their generators. Another application is anew construction of free subgroups in the unit group. In all the constructions dealt with, pairs ofsubgroups ( H, K ), called strong Shoda pairs, and explicit constructed central elements e ( G, H, K )play a crucial role. For arbitrary finite groups we prove that the primitive central idempotentsof the rational group algebras are rational linear combinations of such e ( G, H, K ), with (
H, K )strong Shoda pairs in subgroups of G . Introduction
The investigation of the unit group U ( Z G ) of the integral group ring Z G of a finite group G hasa long history and goes back to work of Higman [Hig] and Brauer [Bra]. One of the reasons forthe importance of the integral group ring Z G is that it is an algebraic tool that links group andring theory. It was anticipated for a long time that the defining group G would be determined byits integral group ring, i.e. if Z G is isomorphic with Z H for some finite group H then G ∼ = H ,the isomorphism problem. Roggenkamp and Scott [RS] showed that this indeed is the case if G isa nilpotent group. Weiss proved a more general result [Wei], which also confirmed a Zassenhausconjecture. It was a surprise when Hertweck [Her] gave a counter example to the isomorphismproblem. In all these investigations the unit group U ( Z G ) of Z G is of fundamental importance.There is a vast literature on the topic. For a survey up to 1994, the reader is referred to the books ofPassman and Sehgal [Pas, Seh1, Seh2]. Amongst many others, during the past 15 years, the followingproblems have received a lot of attention (we include some guiding references): construction ofgenerators for a subgroup of finite index in U ( Z G ) [RS1, RS2, JL1], construction of free subgroups[HP, MS2], structure theorems for U ( Z G ) for some classes of groups G [JPdRRZ].Essential in these investigations is to consider Z G as a Z -order in the rational group algebra Q G and to have a detailed understanding of the Wedderburn decomposition of Q G . To do so, afirst important step is to calculate the primitive central idempotents of Q G . A classical method forthis is to apply Galois descent on the primitive central idempotents of the complex group algebra Mathematics Subject Classification. C
05, 16 S
34, 16 U Key words and phrases.
Idempotents, Group algebras, Group rings, Units.Research partially supported by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk On-derzoek (Flanders), the grant PN-II-ID-PCE-2007-1 project ID 532, contract no. 29/28.09.2007, Ministerio de Cienciay Tecnolog´ıa of Spain and Fundaci´on S´eneca of Murcia. C G . The latter idempotents are the elements of the form e ( χ ) = χ (1) | G | P g ∈ G χ ( g − ) g , where χ runsthrough the irreducible (complex) characters χ of G . Hence the primitive central idempotents of Q G are the elements of the form P σ ∈ Gal( Q ( χ ) / Q ) e ( σ ◦ χ ) (see for example [Yam]). Rather recently,Olivieri, del R´ıo and Sim´on [OdRS1] obtained a character free method to describe the primitivecentral idempotents of Q G provided G is a monomial group, that is, every irreducible character of G is induced from a linear character of a subgroup of G . The new method relies on a theorem of Shodaon pairs of subgroups ( H, K ) of G with K normal in H , H/K abelian and so that an irreduciblelinear character of H with kernel K induces an irreducible character of G . Such pairs are calledShoda pairs of G . In Section 2 we recall the necessary background and explain the description ofthe primitive central idempotents of Q G . It turns out that these idempotents can be built usingthe central elements e ( G, H, K ) (see Section 2 for the definition) with (
H, K ) a Shoda pair of G . Incase the Shoda pair satisfies some extra conditions (one calls such a pair a strong Shoda pair) thenone also obtains a detailed description of the Wedderburn component associated with the centralidempotent. This is an important second step towards a description of the simple components of Q G . This method is applicable to all abelian-by-supersolvable finite groups, in particular to finitenilpotent groups.For arbitrary finite groups G , it remains an open problem to give a character free description ofthe primitive central idempotents of Q G . Only for very few groups that are not monomial, such adescription has been obtained (see for example [GJ] for alternating groups). In section 3 we showthat for arbitrary finite groups G the elements e ( G, H, K ) are building blocks for the constructionof the primitive central idempotents e of Q G , i.e. every such e is a rational linear combination of e ( G, H, K ), where (
H, K ) runs through strong Shoda pairs in subgroups of G . The proof makesfundamental use of Brauer’s Theorem on Induced Characters. Presently we are unable to controlthe rational coefficients in this linear combination.In case G is an abelian-by-supersolvable finite group, then, as mentioned above, the primitivecentral idempotents of Q G are of the form e ( G, H, K ), with (
H, K ) a strong Shoda pair of G andthe simple component Q Ge ( G, H, K ) is described. For nilpotent groups G we will show to have amuch better and detailed control. Indeed, in Section 4 we describe a complete set of matrix units(in particular, a complete set of orthogonal primitive idempotents) of Q Ge ( G, H, K ); a third stepin the description of Q G . This allows us to give concrete representations of the projections ge ofthe group elements g ∈ G as matrices over division rings. As a consequence, the recognition of the Z -order Z G in the Wedderburn description of Q G is reduced to a linear algebra problem over theintegers. We include some examples to show that the method can not be extended to, for example,finite metacyclic groups. It remains a challenge to construct a complete set of primitive idempotentsfor such groups.In Section 5, we give several applications to the unit group U ( Z G ) for G a finite nilpotent group.First we show that if G is a finite nilpotent group such that Q G has no exceptional components(see Section 5 for the definition) then U ( Z G ) has a subgroup of finite index that is generated bythree nilpotent finitely generated groups of which we give explicit generators. The problem ofdescribing explicitly a finite set of generators for a subgroup of finite index in U ( Z G ) has beeninvestigated in a long series of papers. Bass and Milnor did this for abelian groups [Bas], thecase of nilpotent groups so that their rational group algebra has no exceptional components wasdone by Ritter and Sehgal [RS1, RS2], arbitrary finite groups so that their rational group algebrahas no exceptional components were dealt with by Jespers-Leal [JL1]. It was shown that theBass cyclic units together with the bicyclic units generate a subgroup of finite index. Some caseswith exceptional components have also been considered, see for example [GS, Jes, JL2, Seh3]). In ATIONAL GROUP ALGEBRAS OF FINITE GROUPS 3 general, very little is known on the structure of the group generated by the Bass cyclic units and thebicyclic units, except that “often” two of them generate a free group of rank two (see for example[GP, GdR, Jes, JdRR, MS2, JL3]). In this paper, for G a finite nilpotent group, we not only givenew generators for a subgroup of finite index, but more importantly, the generating set is dividedinto three subsets, one of them generating a subgroup of finite index in the central units and eachof the other two generates a nilpotent group. One other advantage of our method with respect tothe proofs and results given in [JL1, RS1, RS2] is that our proofs are (modulo the central units)more direct and constructive to obtain an explicit set of generators for a subgroup of finite indexin the group of units of the integral group ring of a finite nilpotent group. Furthermore, we alsogive new explicit constructions of free subgroups of rank two.2. Preliminaries
We introduce some useful notation and results, mainly from [JLP] and [OdRS1]. Throughout G is a finite group. If H is a subgroup of a group G , then let “ H = | H | P h ∈ H h ∈ Q G . For g ∈ G , let b g = ” h g i and for non-trivial G , let ε ( G ) = Q (1 − c M ), where M runs through the set of all minimalnormal nontrivial subgroups of G . Clearly, “ H is an idempotent of Q G which is central if and onlyif H is normal in G . If K ⊳ H ≤ G then let ε ( H, K ) = Y M/K ∈M ( H/K ) ( “ K − c M ) , where M ( H/K ) denotes the set of all minimal normal subgroups of
H/K . We extend this nota-tion by setting ε ( K, K ) = “ K . Clearly ε ( H, K ) is an idempotent of the group algebra Q G . Let e ( G, H, K ) be the sum of the distinct G -conjugates of ε ( H, K ), that is, if T is a right transversalof Cen G ( ε ( H, K )) in G , then e ( G, H, K ) = X t ∈ T ε ( H, K ) t , where α g = g − αg for α ∈ C G and g ∈ G . Clearly, e ( G, H, K ) is a central element of Q G . If the G -conjugates of ε ( H, K ) are orthogonal, then e ( G, H, K ) is a central idempotent of Q G .A Shoda pair of a finite group G is a pair ( H, K ) of subgroups of G with the properties that K E H , H/K is cyclic, and if g ∈ G and [ H, g ] ∩ H ⊆ K then g ∈ H . A strong Shoda pair of G is a Shoda pair ( H, K ) of G such that H E N G ( K ) and the different conjugates of ε ( H, K ) areorthogonal. We also have, in this case, that Cen G ( ε ( H, K )) = N G ( K ) and H/K is a maximalabelian subgroup of N G ( K ) /K [OdRS1].If χ is a monomial character of G then χ = ψ G , the induced character of a linear character ψ of a subgroup H of G . By a Theorem of Shoda, a monomial character χ = ψ G as above isirreducible if and only if ( H, Ker ψ ) is a Shoda pair (see [Sho] or [CR, Corollary 45.4]). A finitegroup G is monomial if every irreducible character of G is monomial and it is strongly monomialif every irreducible character of G is strongly monomial. It is well known that every abelian-by-supersolvable group is monomial (see [Hup, Theorem 24.3]) and in [OdRS1] it is proved that it iseven strongly monomial. We will use these results in order to study the primitive idempotents ofgroup algebras for some abelian-by-supersolvable groups, including finite nilpotent groups. We willalso use the following description of the simple component associated to a strong Shoda pair. Theorem 2.1. [OdRS1, Proposition 3.4] If ( H, K ) is a strong Shoda pair of G then Q Ge ( G, H, K ) ∼ = M r ( Q N ε ( H, K )) ∼ = M r ( Q ( ζ m ) ∗ ατ N/H ) , ERIC JESPERS, GABRIELA OLTEANU, AND ´ANGEL DEL R´IO where m = [ H : K ] , N = N G ( K ) , r = [ G : N ] and the action α and twisting τ are given by α ( nH )( ζ m ) = ζ im , τ ( nH, n ′ H ) = ζ jm , if n − hnK = h i K and [ n, n ′ ] K = h j K , for hK a generator of H/K , n, n ′ ∈ N and i, j ∈ Z . In the above theorem, ζ m denotes a primitive m -th root of unity and we have used the notation L ∗ ατ G , for L a field and G a group, to denote a crossed product with action α : G → Aut( L ) andtwisting τ : G × G → L ∗ [Pas], i.e. L ∗ ατ G is the associative ring L g ∈ G Lu g with multiplicationgiven by the following rules: u g a = α g ( a ) u g , u g u h = τ ( g, h ) u gh . If the action of G on L is faithful then one may identify G with a group of automorphisms of L and the center of L ∗ ατ G is the fixed subfield F = L G , so that G = Gal(L / F), and this crossedproduct is usually denoted by (
L/F, τ ) [Rei]. We refer to these crossed products as classical crossedproducts. This is the case for the crossed product Q N ε ( H, K )) ∼ = Q ( ζ m ) ∗ ατ N/H in Theorem 2.1which can be described as ( Q ( ζ m ) /F, τ ), where F is the center of the algebra, which is determinedby the Galois action given in Theorem 2.1.3. Primitive central idempotents
For an irreducible character χ of G and a field F of characteristic 0, e F ( χ ) denotes the onlyprimitive central idempotent of F G such that χ ( e ) = 0. In this section, using Brauer’s Theorem onInduced Characters, we give a description of every primitive central idempotent e Q ( χ ) of a rationalgroup algebra Q G corresponding to an irreducible character χ of a finite group G as a rationallinear combination of elements of the form e ( G, H i , K i ), with each ( H i , K i ) a strong Shoda pair ina subgroup of G , or equivalently, K i is a normal subgroup of H i with H i /K i cyclic. Theorem 3.1 (Brauer) . [Bra] Every complex character χ of a finite group G is a Z –linear combina-tion χ = P i a i θ Gi , a i ∈ Z , of characters induced from linear characters θ i of elementary subgroups M i of G , where by an elementary subgroup of G we mean one which is a direct product of a cyclicgroup and a p -group for some prime p . In particular, the M i ’s are cyclic–by– p i -groups for some primes p i , hence by [OdRS1] each M i is strongly monomial. As a consequence, every irreducible character of such a subgroup M i is aninduced character θ M i i from a linear character θ i of a subgroup H i of M i . So, θ M i i is irreducible and( H i , ker( θ i )) is a strong Shoda pair of M i .We also will use the result [OdRS1, Theorem 2.1.] that describes the primitive central idempo-tents e Q ( ψ G ) of a rational group algebra Q G associated to a monomial irreducible character ψ G as e Q ( ψ G ) = [Cen G ( ε ( H, K )) : H ][ Q ( ψ ) : Q ( ψ G )] e ( G, H, K )(1)where ψ is a linear character of the subgroup H of G and K is the kernel of ψ . Proposition 3.2.
Let G be a finite group of order n and χ an irreducible character of G . Thenthe primitive central idempotent e Q ( χ ) of Q G associated to χ is of the form e Q ( χ ) = 1[ Q ( ζ n ) : Q ( χ )] X i a i [ Q ( ζ n ) : Q ( ψ i )][ C i : H i ] e ( G, H i , K i ) ATIONAL GROUP ALGEBRAS OF FINITE GROUPS 5 where a i ∈ Z , ( H i , K i ) are strong Shoda pairs of subgroups of G (equivalently H i /K i is a cyclicsection of G ), C i = Cen G ( ε ( H i , K i )) and ψ i are linear characters of H i with kernel K i .Proof. As it was mentioned in the introduction, for every χ ∈ Irr( G ), we have e Q ( χ ) = X σ ∈ Gal( Q ( χ ) / Q ) e ( χ σ ) = X σ ∈ Gal( Q ( χ ) / Q ) σ ( e ( χ )) = tr Q ( χ ) / Q ( e ( χ )) , where χ σ is the character of G given by χ σ ( g ) = σ ( χ ( g )), for g ∈ G . The interpretation of e Q ( χ )as a trace, suggests the following useful notation for the next arguments. For any finite Galoisextension F of Q containing Q ( χ ), let e F Q = X σ ∈ Gal( F/ Q ) σ ( e ( χ )) = tr F/ Q ( e ( χ )) . Hence e Q ( χ ) = e Q ( χ ) Q ( χ ) = F : Q ( χ )] ( e F Q ( χ )) for every finite Galois extension F of Q ( χ ). UsingBrauer’s Theorem on Induced Characters, we now may write χ = P i a i ψ Gi , with ψ i linear charactersof elementary subgroups H i with kernel K i and a i ∈ Z . Then e Q ( ζ n ) Q ( χ ) = X i a i e Q ( ζ n ) Q ( ψ Gi )and, for every i, we will compute e Q ( ζ n ) Q ( ψ Gi ), as in the proof of [OdRS1, Proposition 2.1.]. (Notethat Q ( ζ n ) contains Q ( χ ), because it is an splitting field of G .)Put e i = e ( ψ i ). We know that A = Aut( C ) acts on the left and G acts on the right on ψ i and on e i (by composition and by conjugation respectively) and that their actions are compatible. Henceone may consider A × G acting on the left on the set of irreducible characters of subgroups of G (and similarly on the e i ’s) by ( σ, g ) · ψ i = σ · ψ i · g − .Let Gal( Q ( ζ n ) / Q ) = { σ , . . . , σ l } and T i = { g , . . . , g m } a right transversal of H i in G . Denoteby C i = Cen G ( ε ( H i , K i )). We have that P mk =1 e i · g k = e ( ψ Gi ), hence e Q ( ζ n ) Q ( ψ Gi ) = l X j =1 σ j e ( ψ Gi ) = l X j =1 m X k =1 σ j e i · g k = m X k =1 tr Q ( ζ n ) / Q ( e i ) · g k = m X k =1 [ Q ( ζ n ) : Q ( ψ i )]tr Q ( ψ i ) / Q ( e i ) · g k = m X k =1 [ Q ( ζ n ) : Q ( ψ i )] ε ( H i , K i ) g k = [ Q ( ζ n ) : Q ( ψ i )][ C i : H i ] e ( G, H i , K i )The above computations now easily yield the desired formula for e Q ( χ ). (cid:3) Remark 3.3.
Notice that the formula from Proposition 3.2 for the computation of the primitivecentral idempotents e Q ( χ ) of Q G associated to an irreducible character χ of G coincides withformula (1) in case χ is a monomial irreducible character of G , that is χ is induced to G from onlyone linear character ψ of a subgroup H , with kernel K such that ( H, K ) is a Shoda pair of G .In general, as seen in Proposition 3.2, one has to consider all strong Shoda pairs in subgroupsof G that contribute to the description of a primitive central idempotent of Q G . However, onecan reduce the search of the Shoda pairs that determine the primitive central idempotents of Q G to representatives given by a relation between such pairs of subgroups. Indeed, in [OdRS2,Proposition 1.4], it is proved that if ( H , K ) and ( H , K ) are two Shoda pairs of a finite group G and α , α ∈ Q are such that e i = α i e ( G, H i , K i ) is a primitive central idempotent of Q G for i = 1 ,
2, then e = e if and only if there is g ∈ G such that H g ∩ K = K g ∩ H . ERIC JESPERS, GABRIELA OLTEANU, AND ´ANGEL DEL R´IO
Remark 3.4.
We would like to be able to give a bound for the integers a i used in the previousproposition and one would also like to give more information on the pairs of groups ( H i , K i ) thatone has to consider in the description of e Q ( χ ).Notice that for monomial (respectively strongly monomial) groups, all primitive central idem-potents are realized as elements of the form αe ( G, H, K ), with α ∈ Q , for some Shoda pair ( H, K )(respectively strong Shoda pair and α = 1) in G . However, for the smallest non-monomial group,which is SL(2 , e = e ( G, B, A ), and e = e ( G, B, − e ( G, B, A ), with G = h x, y i ⋊ h a i , a semidirect product of the quaternion group h x, y i of order 8 by the cyclic group A = h a i of order 3, and with B = h x a i . However, e can notbe written as a rational linear multiple of some e ( G, H, K ) with (
H, K ) a pair of subgroups of G such that K E H . 4. Primitive idempotents for finite nilpotent groups
We start this section by showing a method to produce a complete set of orthogonal primitiveidempotents of a classical crossed product with trivial twisting τ = 1, i.e. τ ( g, h ) = 1, for every g, h ∈ G . Let L be a field of characteristic zero. Observe that ( L/F, ≃ M n ( F ), with n = [ L : F ],therefore a complete set of orthogonal primitive idempotents of ( L/F,
1) contains n idempotents. Lemma 4.1.
Let A = ( L/F, be a classical crossed product with trivial twisting and let G =Gal(L / F) with n = | G | . Let e = | G | P g ∈ G u g and let x , . . . , x n be non-zero elements of L . Thenthe conjugates of e by x , . . . , x n form a complete set of orthogonal primitive idempotents of A ifand only if tr L/F ( x i x − j ) = 0 for every i = j . ( tr L/F is the trace of L over F .)Proof. As the twisting is trivial, { u g : g ∈ G } is a subgroup of order | G | of the group of units of A and hence e is an idempotent of A . Moreover u g e = e for every g ∈ G . Therefore, if x ∈ L then exe = | G | P g ∈ G u g xe = | G | P g ∈ G x g − u g e = | G | P g ∈ G x g e = | G | tr L/F ( x ) e . Thus, if x ∈ L then e and xex − are orthogonal if and only if tr L/F ( x ) = 0 and the lemma follows. (cid:3) Examples 4.2. (1) In the proof of Theorem 4.5 we will encounter some examples of classicalcrossed products with trivial twisting with a list x , . . . , x n satisfying the conditions of theprevious lemma.(2) Another situation where one can find always such elements correspond to the case when L/F is a cyclic extension of order n and F contains an n -root of unity. Then L is thesplitting field over F of an irreducible polynomial of F [ X ] of the form X n − a . If α ∈ L with α n = a then x = 1 , x = α, . . . , x n = α n − satisfy the conditions of Lemma 4.1.Indeed, the minimal polynomial of α i over F for 1 ≤ i < n is of the form X n/d − a i/d for d = gcd( n, i ) and therefore tr L/F ( α i ) = [ L : F ( α i )]tr F ( α i ) /F ( α i ) = 0 and similarlytr L/F ( α − i ) = 0.(3) We now construct an example where there are no elements x , . . . , x n satisfying the con-ditions of Lemma 4.1. Consider the trivial cyclic algebra ( L = Q ( ζ ) /F = Q ( √− ,
1) ofdegree 3. If x , x , x satisfy the conditions of Lemma 4.1 then α = x x − and α − = x x − are non-zero elements of L with zero trace over F . This implies that the minimal polyno-mial of α over F is of the form X − a for some a ∈ F . But this implies that F contains athird root of unity, a contradiction. ATIONAL GROUP ALGEBRAS OF FINITE GROUPS 7
The groups listed in the following lemma will be the building blocks in the proof of Theorem 4.5.For n and p integers with p prime, we use v p ( n ) to denote the valuation at p of n , i.e. p n is themaximum p -th power dividing n . Lemma 4.3.
Let G be a finite p -group which has a maximal abelian subgroup which is cyclic andnormal in G . Then G is isomorphic to one of the groups given by the following presentations: P = ¨ a, b | a p n = b p k = 1 , b − ab = a r ∂ , with either v p ( r −
1) = n − k or p = 2 and r . P = ¨ a, b, c | a n = 1 , b k = 1 , c = 1 , bc = cb, b − ab = a r , c − ac = a − ∂ , with r ≡ . P = ¨ a, b, c | a n = 1 , b k = 1 , c = a n − , bc = cb, b − ab = a r , c − ac = a − ∂ , with r ≡ . Note that if k = 0 (equivalently, if b = 1) then the first case correspond to the case when G is abelian (and hence cyclic), the second case coincides with the first case with p = 2, k = 1 and r = −
1, and the third case is the quaternion group of order 2 n +1 . Proof.
Let A be a maximal abelian subgroup of G and assume that A is cyclic (generated by a )and normal in G . Put | A | = p n . Consider the action of G on A by inner automorphisms. Since A ismaximal abelian in G , the kernel of this action is A and therefore G/A is isomorphic to a subgroupof the group of automorphisms of A .If either p is odd or p = 2 and n ≤ A ) is cyclic and otherwise Aut( A ) = h φ i × h φ − i ,where φ r is the automorphism of A given by φ r ( x ) = x r .Assume that G/A is cyclic, so that G has a presentation of the form(2) G = ¨ a, b | a p n = 1 , b p k = a s , b − ab = a r ∂ , with p n | r p k − p n | s ( r − p i ≥ xp i ) p ≡ xp i +1 mod p i +2 for every i ≥ x ∈ Z . Using this, one deduces that if either p is odd or p = 2 and r ≡ v p ( r p i −
1) = v p ( r p i − − i ≥
1. Furthermore, from the assumption that A is maximalabelian in G , one deduces that n ≤ v p ( r p k −
1) = v p ( r p k − − ≤ n and hence v p ( r p k −
1) = n and v p ( r −
1) = n − k . Therefore, v p ( s ) ≥ n − v p ( r −
1) = k = v p (cid:16) r pk − r − (cid:17) = v p Ä r + r + · · · + r p k − ä and hence there is an integer x such that x (1 + r + r + · · · + r p k − ) + s ≡ p n . Then( a x b ) p k = 1 and, replacing b by a x b in (2), we obtain the presentation of P . We have also provedthat v p ( r −
1) = n − k unless p = 2 and r p = 2 and r v ( s ) ≥ n − v ( r −
1) = n − v (1 + r + r + . . . r k − ) = v (cid:16) r k − r − (cid:17) ≥ n −
1. If v ( s ) ≥ n then G ∼ = P . If v ( s ) = v (1 + r + r + . . . r k − ) = n − ab ) k = 1. Replacing b by ab weobtain again that G ∼ = P . Otherwise, v ( s ) = n − v (1 + r + r + . . . r k − ) = n . Therefore v ( r k − ) > n > v ( r k − −
1) and by the first part of the proof (applied to (cid:10) a, b (cid:11) ) this impliesthat k = 2. Then G is the quaternion group of order 2 n +1 which is P for k = 0.Assume now that G/A is not cyclic, so p = 2 and G/ h a i = h b i × h c i with c acting by inversionon h a i . This provides a presentation of G of the form(3) G = h a, b, c | a n = 1 , b k = a s , ca = a − c, cb = a i bc, b − ab = a r , c = 1 or c = a n − i . Replacing b by bc if needed, one may assume that v ( r −
1) = n − k ≥ v ( r k −
1) = n .So, applying the first part of the proof to h a, b i , we may assume that b k = 1. Then c = b k cb − k = a − i (1+ r + ... + r k − ) c and so 2 n | − i (1 + r + . . . + r k − ) = − i r k − r − . As v (cid:16) r k − r − (cid:17) = k , we have ERIC JESPERS, GABRIELA OLTEANU, AND ´ANGEL DEL R´IO v ( i ) ≥ n − k = v ( r − j so that j ( r − − i ≡ n . Itis easy to verify that the commutator of b and a j c is 1. So, replacing c by a j c if needed, we mayassume that b and c commute and we obtain the presentation of P , if c = 1, and the presentationof P , if c = a n − . (cid:3) We also need the following result on splitting of a Hamiltonian quaternion algebra H ( F ) = F [ i, j | i = j = − , ji + ji = 0]. Lemma 4.4.
Let F be a field of characteristic zero. Then the quaternion algebra H ( F ) splits ifand only if x + y = − for some x, y ∈ F . In that case (1 + xi + yj ) and (1 − xi − yj ) form acomplete set of primitive idempotents of H ( F ) .Furthermore, if F = Q ( ζ m , ζ n + ζ − n ) with m odd then − is the sum of two squares of F if andonly if m = 1 and either n ≥ or the multiplicative order of modulo m is even.Proof. The first part can be found in [Seh1, Proposition 1.13]. Now assume that F = Q ( ζ m , ζ n + ζ − n ) with m odd. If m = 1 then F is totally real and therefore − F . So assume that m = 1. If n ≤
2, then F = Q ( ζ m ) and the result is well known (see for example[Mos, FGS] or [Lam, pages 307–308]). Finally assume that m = 1 and n ≥
3. Then F contains √ Q , the duadic completion of Q [Lam, Corollary 2.24], we deduce that[ F : Q ] is even. Then − F p for every place p of F and hence − F (see [Lam, page 304]). (cid:3) Now we are ready to show an effective method to calculate a complete set of orthogonal primitiveidempotents of Q G for G a finite nilpotent group. Since G is abelian-by-supersolvable and hencestrongly monomial, it follows from [OdRS1, Theorem 4.4] that every primitive central idempotentof Q G is of the form e ( G, H, K ) with (
H, K ) a strong Shoda pair of G and therefore it is enough toobtain a complete set of orthogonal primitive idempotents of Q Ge ( G, H, K ) for every strong Shodapair (
H, K ) of G . This is described in our main result that we state now. Theorem 4.5.
Let G be a finite nilpotent group and ( H, K ) a strong Shoda pair of G . Set e = e ( G, H, K ) , ε = ε ( H, K ) , H/K = h a i , N = N G ( K ) and let N /K and H /K = h a i (respectively N ′ /K and H ′ /K = h a ′ i ) denote the -parts (respectively, ′ -parts) of N/K and
H/K respectively.Then h a ′ i has a cyclic complement h b ′ i in N ′ /K .A complete set of orthogonal primitive idempotents of Q Ge consists of the conjugates of c b ′ β ε bythe elements of T ′ T T G/N , where T ′ = { , a ′ , a ′ , . . . , a [ N ′ : H ′ ] − ′ } , T G/N denotes a left transver-sal of N in G and β and T are given according to the cases below. (1) If H /K has a complement M /K in N /K then β = ” M . Moreover, if M /K is cyclicthen there exists b ∈ N such that N /K is given by the following presentation ≠ a , b | a n = b k = 1 , a b = a r ∑ , and if M /K is not cyclic, there exist b , c ∈ N such that N /K is given by the followingpresentation ≠ a , b , c | a n = b k = 1 , c = 1 , a b = a r , a c = a − , [ b , c ] = 1 ∑ , with r ≡ (or equivalently, a n − is central in N /K ). Then (i) T = { , a , a , . . . , a k − } , if a n − is central in N /K and M /K is cyclic; and ATIONAL GROUP ALGEBRAS OF FINITE GROUPS 9 (ii) T = { , a , a , . . . , a k − − , a n − , a n − +12 , . . . , a n − +2 k − − } , otherwise. (2) if H /K has no complement in N /K then there exist b , c ∈ N such that N /K is givenby the following presentation ≠ a , b , c | a n = b k = 1 , c = a n − , a b = a r , a c = a − , [ b , c ] = 1 ∑ , with r ≡ and we set m = [ H ′ : K ] / [ N ′ : H ′ ] . Then (i) β = “ b and T = { , a , a , . . . , a k − } , if either H ′ = K or the order of modulo m is odd and n − k ≤ and (ii) β = “ b xa n − + ya n − c and T = { , a , a , . . . , a k − , c , a c , a c , . . . , a k − c } with x, y ∈ Q î a [ N ′ : H ′ ]2 ′ , a k + a − k ó , satisfying (1 + x + y ) ε = 0 , if H ′ = K and either the order of modulo m is evenor n − k > .Proof. We start the proof by making some useful reductions. Taking T = T G/N a left transversalof N in G , the conjugates of ε by elements of T are the “diagonal” elements in the matrix algebra Q Ge = M G/N ( Q N ε ). Hence, following the proof of [OdRS1, Proposition 3.4], one can see that it issufficient to compute a complete set of orthogonal primitive idempotents for Q N ε = Q H ε ∗ N/H and then add their T -conjugates in order to obtain the primitive idempotents of Q Ge . So one mayassume that N = G , i.e. K is normal in G and hence e = ε and T = { } . Then the naturalisomorphism Q G “ K ≃ Q ( G/K ) maps ε to ε ( H/K ). So, from now on we assume that K = 1 andhence H = h a i is a cyclic maximal abelian subgroup of G , which is normal in G and e = ε = ε ( H ).If G = H then Q Ge is a field, T = T ′ = { } and b ′ = β = 1; hence the result follows. So, in theremainder of the proof we assume that G = H .The map aε ζ induces an isomorphism f : Q Hε → Q ( ζ ), where ζ is a primitive | H | -root ofunity. Using the description of Q Ge given in Theorem 2.1, one obtains a description of Q Ge as aclassical crossed product ( Q ( ζ ) /F, τ ), where F is the image under f of the center of Q Ge .We first consider the case when G is a p -group. Then G and H = h a i satisfy the conditionsof Lemma 4.3 and therefore G is isomorphic to one of the three groups of this lemma. Moreover, H has a complement in G if and only if G ∼ = P or G ∼ = P and, in these cases, τ is trivial. Weclaim that in these cases it is possible to give a list of elements x , . . . , x p k of Q ( ζ ) ( p k = [ G : H ])satisfying the conditions of Lemma 4.1 and the elements f − ( x ) , . . . , f − ( x p k ) correspond to theconjugating elements in G given in the statement of the theorem in the different cases. To provethis we will use the following fact: if K is a subfield of Q ( ζ ) such that ζ p ∈ K , ζ i K (with i = 1 , . . . , p k −
1) and, moreover, ζ ∈ K if p = 2 then tr Q ( ζ ) /K ( ζ i ) = 0. To see this notice thatif d is the minimum integer such that ζ ip d ∈ K then Q ( ζ i ) /K is cyclic of degree p d and ζ i is aroot of X p d − ζ ip d ∈ K [ X ]. Then X p d − ζ ip d is the minimal polynomial of ζ i over K . Hencetr Q ( ζ i ) /K ( ζ i ) = 0 and thus tr Q ( ζ ) /K ( ζ i ) = 0.Assume first that G = P and v p ( r −
1) = n − k (equivalently a p n − k p ∈ Z ( G )), that is, either p is oddor p = 2 and r ≡ F is the unique subfield of index [ G : H ] = p k in Q ( ζ ) and suchthat if p = 2 then ζ ∈ F . Namely F = Q ( ζ p n − k ) = Q ( ζ p k ). If we set x i = ζ i , for i = 0 , , . . . , p k − x i x − j = ζ i − j . If i = j then ζ i − j F and hence tr Q ( ζ ) /F ( x i x − j ) = tr Q ( ζ ) /F ( ζ i − j ) = 0. Thus,by Lemma 4.1, the conjugates of b b by 1 , ζ, ζ , . . . , ζ p k − form a complete set of orthogonal primitive idempotents of ( Q ( ζ ) /F, f − ( x i ) form the elements of T ′ if p is odd or theelements of T , in case (1.i).Assume now that G is still P , but with p = 2 and r a n − is notcentral). In this case ζ F and F ( ζ ) is the unique subextension of Q ( ζ ) / Q ( ζ ) of index [ G : H ] / k − . That is F ( ζ ) = Q ( ζ n − k +1 ) = Q ( ζ k − ). We take x i = ζ i and x k − + i = ζ n − + i = ζ ζ i , for 0 ≤ i < k − . Hence, if i = j then x i x − j is either ζ ± or ζ ± i or ζ ± ζ ± i , with i =1 , , . . . , k − −
1. As ζ i F ( ζ ), we have tr Q ( ζ ) /F ( ζ ) ( ζ i ) = 0. Sincetr Q ( ζ ) /F ( ζ ) = tr F ( ζ ) /F tr Q ( ζ ) /F ( ζ ) ( ζ ) = [ Q ( ζ ) : Q ( ζ )]tr F ( ζ ) /F ( ζ ) = 0 , tr Q ( ζ ) /F ( ζ i ) = tr F ( ζ ) /F tr Q ( ζ ) /F ( ζ ) ( ζ i ) = 0and tr Q ( ζ ) /F ( ζ ζ i ) = tr F ( ζ ) /F tr Q ( ζ ) /F ( ζ ) ( ζ ζ i ) = tr F ( ζ ) /F ( ζ tr Q ( ζ ) /F ( ζ ) ( ζ i )) = 0 , we deduce that tr( x i x − j ) = 0 for every i = j . Then f − maps these elements to the elements of T for case (1.ii).Now assume that G = P . Then F = Q ( ζ n − k + ζ − n − k ). Since r ≡ n − k ≥
2. Thenthe same argument as in the previous case shows that the 2 k +1 elements of the form x i = ζ i and x k + i = ζ n/ + i = ζ ζ i , for 0 ≤ i < k satisfy the conditions of Lemma 4.1. The elements f − ( x i )form now the set T of case (1.ii).Now we consider the non-splitting case, i.e. G = P . Then the center of Q Ge is isomorphic to F = Q ( ζ ) h b,c i = Q ( ζ n − k + ζ − n − k ) and b b Q G ε b b = b b Q h a, c i ε b bF + F a n − + F c + F ( a n − c ) ∼ = H ( F ),which is a division algebra, as F is a real field. Then b bε is a primitive idempotent of Q Ge . Hence Q Ge ≃ M k ( H ( F )) and from the first case one can provide the 2 k orthogonal primitive idempotentsneeded in this case by taking the conjugates of b b by 1 , a, a , . . . , a k − , and this agrees with case(2.i). This finishes the p -group case.Let us now consider the general case, where G is not necessarily a p -group. Then G = G × G p ×· · ·× G p r = G × G ′ , with p i an odd prime for every i = 1 , . . . , r . Then ( H,
1) is a strong Shoda pairof G if and only if ( H p i ,
1) is a strong Shoda pair of G p i , for every i = 0 , , . . . , m , (with p = 2) and ε ( H ) = Q i ε ( H p i ). Using this and a dimension argument it easily follows that the simple algebra Q Gε ( H ) is the tensor product over Q of the simple algebras Q G p i ε ( H p i ). On the other hand, wehave seen that for i ≥ Q G p i ε ( H i ) ≃ M p kii ( Q ( ζ p ni − kii )), for p n i i = | H i | and p k i i = [ N i : K i ]. Then Q G ′ ε ( H ′ ) ≃ M [ G ′ : H ′ ] ( Q ( ζ m )), with m = | H ′ | / [ G ′ : H ′ ] (= [ H ′ : K ] / [ G ′ : H ′ ]) and thena complete set of orthogonal primitive idempotents of Q G ′ ε ( H ′ ) can be obtained by multiplyingthe different sets of idempotents obtained for each tensor factor. Observe that each G p i , with i ≥ h a i i ⋊ h b i i and so G ′ = h a i ⋊ h b i , with a = a . . . a r and b = b . . . b r . Having inmind that a p kii is central one can easily deduce, with the help of the Chinese Remainder Theorem,that the product of the different primitive idempotents of the factors from the odd part (i.e. theconjugates of b b i by 1 , a i , a i , . . . , a p kii − i are the conjugates of b bε by 1 , a, a , . . . , a [ G ′ : H ′ ] − . In thenotation of the statement of the theorem, a = a ′ and T ′ = { , a, a , ...., a [ G ′ : H ′ ] − as wanted.If | G | is odd then the proof is finished. Otherwise we should combine the odd and even partsof G . If H has a complement in G then Q G ε ( H ) is split over its center and hence we can take T as in the 2-group case. However, if H does not have a complement in G then Q G ε ( H ) = M [ G : H ] / ( H ( Q ( ζ n − k + ζ − n − k ))) and hence Q Gε = M [ G : H ] / ( H ( F )), with F = Q ( ζ m , ζ n − k + ζ − n − k )).If H ( F ) is not split (equivalently the conditions of (2.i) hold) then we can also take T as in the2-group case. However, if H ( F ) is split then one should duplicate the number of idempotents, or ATIONAL GROUP ALGEBRAS OF FINITE GROUPS 11 equivalently duplicate the size of T . In this case − F . Observing that f ( a [ N ′ : H ′ ]2 ′ ) is a primitive m -th root of unity and f ( a k ) is a primitive 2 n − k root of unity, we deducethat there are x, y ∈ Q ( a [ N ′ : H ′ ]2 ′ , a k + a − k ) such that (1 + x + y ) ε = 0. Then we can duplicatethe number of idempotents by multiplying the above idempotents by f = xa n − + ya n − c and1 − f = − xa n − − ya n − c (see Lemma 4.4). Observing that 1 − f = f c , we obtain that theseidempotents are the conjugates of “ b f ε by 1 , a , . . . , a k − , c , a c , . . . , a k − c , as desired. (cid:3) Remark 4.6.
A description of the simple algebras Q Ge using Theorem 2.1 can be given according tothe cases listed above. Thus, Q Ge = M | G/H | ( Q ( ζ [ H : K ] ) N/K ), that is a matrix algebra over the fixedfield of the natural action of
N/K on the cyclotomic field Q ( ζ [ H : K ] ) = Q Hε ( H, K ), in cases (1) and(2.ii) of Theorem 4.5 and Q Ge = M | G/H | ( H ( Q ( ζ [ H : K ] ) N/K )) in case (2.i). In particular, if Q Ge isa non-commutative division algebra then [ G : H ] = 2, N = G and Q Ge ∼ = H Ä Q Ä ζ [ H : K ] + ζ − H : K ] ää ,a totally definite quaternion algebra.As a consequence of Theorem 4.5, we get the following result on the Schur indices of the simplecomponents of group algebras for finite nilpotent groups over fields of characteristic zero. Theorem 4.7 (Roquette) . Let G be a finite nilpotent group and F a field of characteristic zero.Then F G ∼ = L i M n i ( D i ) , where D i are quaternion division algebras if not commutative, that is theSchur index of the simple components of F G is at most . If the Schur index of a simple componentof F G is then the Sylow -subgroup of G has a quaternion section. Remark 4.8.
Notice that the use of Lemma 4.1 has been essential in all the cases of the proof ofTheorem 4.5. We would like to be able to give a similar description to the one from Theorem 4.5for a complete set of orthogonal primitive idempotents for rational group algebras of arbitrary finitemetacyclic groups. Unfortunately, the approach of Theorem 4.5 does not apply here. For example,if G = C ⋊ C = h a i ⋊ h b i , with b − ab = a and ε = ε ( h a i ) then there is not a complete set oforthogonal primitive idempotents of Q Gε formed by Q ( a )-conjugates of b bε . This is a consequenceof Example 4.2 (3). Notation 4.9.
As an application of Theorem 4.5 we next will describe a complete set of matrixunits in a simple component Q Ge , where e = e ( G, H, K ) with (
H, K ) a strong Shoda pair ofa finite nilpotent group G . A complete set of primitive idempotents of Q Ge ( G, H, K ) is givenaccording to the cases of Theorem 4.5. Using the notation in these different cases of Theorem4.5, let T e = T ′ T T G/N and β e = c b ′ β ε , where ε = ε ( H, K ), T G/N denotes a left transversal of N = N G ( K ) in G ; T ′ = ¶ , a ′ , . . . , a [ N ′ : H ′ ] − ′ © ; T = ¶ , a , . . . , a k − © , in cases (1.i) and (2.i); ¶ , a , . . . , a k − − , a n − , a n − +12 , . . . , a n − +2 k − − © , in case (1.ii); ¶ , a , . . . , a k − , c , a c , . . . , a k − c © , in case (2.ii);and β = ” M , in case (1); “ b , in case (2.i); “ b xa n − + ya n − c , in case (2.ii). Corollary 4.10.
Let G be a finite nilpotent group. For every primitive central idempotent e = e ( G, H, K ) , with ( H, K ) a strong Shoda pair of G , let T e and β e be as in Notation 4.9. For every t, t ′ ∈ T e let E tt ′ = t − β e t ′ . Then { E tt ′ | t, t ′ ∈ T e } gives a complete set of matrix units in Q Ge , i.e. e = P t ∈ T e E tt and E t t E t t = δ t t E t t , for every t , t , t , t ∈ T e .Moreover, E tt Q GE tt ∼ = F , in cases (1) and (2.ii) of Theorem 4.5, and E tt Q GE tt = H ( F ) , incase (2.i) of Theorem 4.5, where F is the fixed subfield of Q ( a ) ε under the natural action of N/H .Proof.
We know from Theorem 4.5 that the set { E tt | t ∈ T e } is a complete set of primitiveidempotents of Q Ge . From the definition of the E tt ′ it easily follows that E t t E t t = δ t t E t t ,for t i ∈ T e , i = 1 , . . . ,
4. The second statement is already mentioned in Remark 4.6. (cid:3) Generators of a subgroup of finite index in U ( Z G )As an application of the description of the primitive central idempotents in Corollary 4.10, wenow can easily explicitly construct two nilpotent subgroups of U ( Z G ) (which correspond with upperrespectively lower triangular matrices in the simple components). Together with the central unitsthey generate a subgroup of finite index in the unit group. We begin by recalling a result of Jespers,Parmenter and Sehgal that gives explicit generators for a subgroup of finite index in the center.First we recall the construction of units known as Bass cyclic units in the integral group ring Z G of a finite group G . Let g ∈ G and suppose g has order n . Let k be an integer so that 1 < k < n and ( k, n ) = 1. Then, k ϕ ( n ) ≡ n , where ϕ is the Euler ϕ -function, and b ( g, k ) = k − X j =0 g j ! ϕ ( n ) + (1 − k ϕ ( n ) ) b g is a unit in Z G . (Note that our notation slightly differs from the one used in [Seh1]; this becauseof our definition of b g = n P ni =1 g i .) The group generated by all Bass cyclic units of Z G we denoteby B ( G ).We now introduce the following units defined in [JPS]. Let Z v denote the v -th center of G, andsuppose that G is nilpotent of class n . For any x ∈ G and b ∈ Z h x i , put b (1) = b and, for 2 ≤ v ≤ n ,put b ( v ) = Y g ∈ Z v b g ( v − . Note that by induction b ( v ) is central in Z h Z v , x i and is independent of the order of the conjugatesin the product expression. In particular, b ( n ) ∈ Z ( U ( Z G )). Let B ( n ) ( G ) = (cid:10) b ( n ) | b a Bass cyclic of Z G (cid:11) . Proposition 5.1. [JPS, Proposition 2] If G is a finite nilpotent group of class n , then B ( n ) ( G ) hasfinite index in Z ( U ( Z G )) . The proof of the previous proposition relies on results of Bass [Bas, Lemma 2.2, Lemma 3.6,Theorem 2, Theorem 4], which states that the natural images of the Bass cyclic units in theWhitehead group K ( Z G ) of Z G generate a subgroup of finite index, and on the fact that thetorsion-free rank of the abelian groups K ( Z G ) and Z ( U ( Z G )) are the same. Clearly, because K ( Z G ) is commutative, the natural image of a b ( n ) in K ( Z G ) is equal with the natural imageof some power of b in K ( Z G ). Hence in K ( Z G ) the group generated by the Bass cyclic units ATIONAL GROUP ALGEBRAS OF FINITE GROUPS 13 contains the group generated by the b ( n ) ’s as a subgroup of finite index. So, indeed, B ( n ) ( G ) is offinite index in Z ( U ( Z G )).If G is a finite group and e is a primitive central idempotent of the rational group algebra Q G ,then the simple algebra Q Ge is identified with M n ( D ), a matrix algebra over a division algebra D . As in [JL1], an exceptional component of Q G is a non-commutative division algebra other thana totally definite quaternion algebra, or a two-by-two matrix algebra over either the rationals, aquadratic imaginary extension of the rationals or a non-commutative division algebra.Let O be an order in D and denote by GL n ( O ) the group of invertible matrices in M n ( O ) and bySL n ( O ) its subgroup consisting of matrices of reduced norm 1. For an ideal Q of O we denote by E ( Q ) the subgroup of SL n ( O ) generated by all Q -elementary matrices, that is E ( Q ) = h I + qE ij | q ∈ Q, ≤ i, j ≤ n, i = j, E ij a matrix unit i . We recall the following celebrated theorem (see forinstance [RS1] or [JL1, Theorem 2.2]). Theorem 5.2 (Bass-Milnor-Serre-Vaserstein) . If n ≥ then [SL n ( O ) : E ( Q )] < ∞ . If n = 2 and D is an algebraic number field which is not rational or imaginary quadratic, then [SL ( O ) : E ( Q )] < ∞ . Because of the description of the matrix units from Corollary 4.10, we can now show that, in case G is nilpotent and does not have exceptional simple components, U ( Z G ) has a subgroup of finiteindex that is generated by three nilpotent groups, one of which is a central subgroup contained inthe group generated by the Bass cyclic units and the others are generated by the units of the form1 + E tt gE t ′ t ′ , with g ∈ G and t, t ′ ∈ T e , with T e as in Notation 4.9. Theorem 5.3.
Let G be a finite nilpotent group of class n such that Q G has no exceptional com-ponents. For every primitive central idempotent e = e ( G, H, K ) , with ( H, K ) a strong Shoda pairof G , let N = N G ( K ) , ε = ε ( H, K ) and let T e and β e be as in Notation 4.9. Fix an order < in T e .Let H/K = h a i and let l be the least integer such that a l ε is central in Q N ε . Then the followingtwo groups are nilpotent subgroups of U ( Z G ) : V + e = (cid:10) | G | t − β e a j t ′ | a j ∈ h a l i , t, t ′ ∈ T e , t ′ > t (cid:11) ,V − e = (cid:10) | G | t − β e a j t ′ | a j ∈ h a l i , t, t ′ ∈ T e , t ′ < t (cid:11) . Hence V + = (cid:10) V + e | e = e ( G, H, K ) a primitive central idempotent of Q G (cid:11) = Y e V + e and V − = (cid:10) V − e | e = e ( G, H, K ) a primitive central idempotent of Q G (cid:11) = Y e V − e are nilpotent subgroups of U ( Z G ) . Furthermore, if B ( n ) ( G ) = h b ( n ) | b a Bass cyclic unit of G i ,where n is the nilpotency class of G , then the group (cid:10) B ( n ) ( G ) , V + , V − (cid:11) is of finite index in U ( Z G ) .Proof. Recall that the intersection of the unit groups of two orders in a finite dimensional rationalalgebra are commensurable and henceforth it is enough to show that (cid:10) B ( n ) , V + , V − (cid:11) contains asubgroup of finite index in the group of units of an order of (1 − e ) Q + Q Ge for a primitive centralidempotent e of Q G . So fix such a primitive central idempotent e = e ( G, H, K ) of Q G .The elements of the form 1 + | G | t − β e a j t ′ , with a j ∈ h a l i and t, t ′ ∈ T e , project triviallyto Q G (1 − e ) and by Corollary 4.10 they project to an elementary matrix of M n ( O ), for some order O in the division ring D , where Q Ge ≃ M n ( D ). Since also | G | β e ∈ Z G , it follows that1 + | G | t − β e a j t ′ ∈ U ( Z G ) and by Theorem 5.2, for t = t ′ , these units generate a subgroup of finiteindex in (1 − e ) + SL n ( O ). By Theorem 5.1, B ( n ) has finite index in Z ( U ( Z G )) and therefore itcontains a subgroup of finite index in the center of (1 − e ) + GL n ( O ). As the center of GL n ( O )together with SL n ( O ) generate a subgroup of finite index of GL n ( O ), we conclude that the group (cid:10) B ( n ) , V + , V − (cid:11) contains a subgroup of finite index in the group of units of an order of (1 − e ) Q + Q Ge .Note that V + e and V − e correspond to upper and lower triangular matrices respectively and hencethey are nilpotent groups. (cid:3) Another application of the construction of the matrix units is that one can easily obtain freesubgroups of U ( Z G ) for G a finite nilpotent group. Corollary 5.4.
Let G be a finite nilpotent group, ( H, K ) a strong Shoda pair of G , ε = ε ( H, K ) , e = e ( G, H, K ) and let T e and β e be as in Notation 4.9. If Q Ge is not a division algebra (seeRemark 4.6), then for every t, t ′ ∈ T e with t = t ′ , (cid:10) | G | t − β e t ′ , | G | t ′− β e t (cid:11) is a free group of rank .Proof. By Corollary 4.10, we may write 1+ | G | t − β e t ′ = 1+ | G | E tt ′ and 1+ | G | t ′− β e t = 1+ | G | E t ′ t .Hence (cid:10) | G | t − β e t ′ , | G | t ′− β e t (cid:11) is isomorphic with ≠Å | G | ã , Å | G | ã∑ . Since | G | ≥
2, a well known result of Sanov yields that this group is a free of rank 2. (cid:3)
Remark 5.5.
A well known result of Hartley and Pickel [HP] (or see for example [Seh1]) saysthat U ( Z G ) contains a free non-abelian subgroup for any finite non-abelian group G that is not aHamiltonian 2-group. Only in 1997, Sehgal and Marciniak in [MS1] gave a concrete constructionof such a group. They showed that if u g,h = 1 + (1 − g ) h b g |h g i| is a non trivial bicyclic unit then ¨ u g,h , u ′ g − h − ∂ is a free group of rank 2. More generally, in [MS2] it is shown that if c ∈ Z G satisfies c = 0 and c = 0 then h c ∗ , c i is free group of rank 2. For c = P g ∈ G z g g we denoteby c ∗ = P g ∈ G z g g − . So, ∗ denotes the classical involution on Z G . The above corollary yields manymore concrete elements that can be substituted for c . Since then, as mentioned in the introduction,there have been several papers on constructing free subgroups in U ( Z G ) generated by Bass and/orbicyclic units. Acknowledgements.
The second author would like to thank for the warm hospitality during thevisit to Vrije Universiteit Brussel with a postdoctoral grant of Fundaci´on S´eneca of Murcia. Theauthors would like to thank Capi Corrales for the help with the splitting of quaternion algebras(Lemma 4.4).
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Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
E-mail address : [email protected] Department of Statistics-Forecasts-Mathematics, Babes¸-Bolyai University, Str. T. Mihali 58-60,400591 Cluj-Napoca, Romania
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