On finite groups acting on spheres and finite subgroups of orthogonal groups
aa r X i v : . [ m a t h . G T ] A ug On finite groups acting on spheresand finite subgroups of orthogonal groups
Bruno P. ZimmermannUniversit`a degli Studi di TriesteDipartimento di Matematica e Informatica34127 Trieste, ItalyAbstract.
This is a survey on old and new results as well as an introduction to vari-ous related basic notions and concepts, based on two talks given at the InternationalWorkshop on Geometry and Analysis in Kemerovo (Sobolev Institute of Mathematics,Kemerovo State University) and at the University of Krasnojarsk in June 2011. Wediscuss finite groups acting on low-dimensional spheres, comparing with the finite sub-groups of the corresponding orthogonal groups, and also finite simple groups acting onspheres and homology spheres of arbitrary dimension.
1. Introduction
We are interested in the class of finite groups which admit an orientation-preservingaction on a sphere S n of a given dimension n . All actions in the present paperwill be faithful and orientation-preserving (but no necessarily free). So formally anaction of a finite group G is an injective homomorphism from G into the group oforientation-preserving homeomorphism of S n ; informally, we will consider G as a groupof orientation-preserving homeomorphisms of S n and distinguish various types of ac-tions: topological actions : G acts by homeomorphisms; smooth actions : G acts by diffeomorphisms; linear (orthogonal) actions : G acts by orthogonal maps of S n ⊂ R n +1 (that is, G is asubgroup of the orthogonal group SO( n + 1)); or, more generally, any topological actionwhich is conjugate to a linear action; locally linear actions : topological actions which are linear in regular neighbourhoods offixed points of any nontrivial subgroup.It is well-known that smooth actions are locally linear (by the existence of equivariantregular neighbourhoods, see [Bre]; see also the discussion in the next two sections forsome examples). 1he reference model for finite group actions on S n is the orientation-preserving orthog-onal group SO( n + 1); in fact, the only examples of actions which can easily be seen arelinear actions by finite subgroups of SO( n + 1). A rough general guiding line is then thefollowing: Motivating naive conjecture : Every action of a finite group on S n is linear (that is,conjugate to a linear action). More generally, how far can an action by homeomorphismsor diffeomorphisms be from a linear action?It is one of the main features of linear actions that fixed point sets of single elements arestandard unknotted spheres S k in S n (that is, intersections of linear subspaces R k +1 of R n +1 with S n ). We note that it is a classical and central result of Smith fixed pointtheory that, for an action of a finite p -group (a group whose order is a power of a prime p ) on a mod p homology n -sphere (a closed n -manifold with the mod p homology of S n ), fixed point sets are again mod p homology spheres (see [Bre]; one has to considerhomology with coefficients in the integers mod p here since this does not remain truein the setting of integer homology spheres).We note that for free actions on S n (nontrivial elements have empty fixed point sets),the class of finite groups occurring is very restricted (they have periodic cohomology, ofperiod n + 1, see [Bro]). On the other hand, for not necessarily free actions, all finitegroups occur for some n (by just considering a faithful, real, linear representation ofthe finite group). Two of the motivating problems of the present survey are then thefollowing: Problems. i) Given a dimension n , determine the finite groups G which admit anaction on a sphere or a homology sphere of dimension n (comparing with the class offinite subgroups of the orthogonal group SO( n + 1)).ii) Given a finite group G , determine the minimal dimension of a sphere or homologysphere on which G admits an action (show that it coincides with the minimal dimensionof a linear action of G on a sphere).We close the introduction with a general result by Dotzel and Hamrick ([DH]), againfor finite p -groups: If G is a finite p -group acting smoothly on a mod p homology n -sphere then G admits also a linear action on S n such that the two actions have thesame dimension function for the fixed point sets of all subgroups of G .In the next sections we will discuss finite groups acting on low-dimensional spheres,starting with the 2-sphere S . 2 . Finite groups acting on S and finite subgroups of SO(3) . Let G be a finite group of (orientation-preserving) homeomorphisms of the 2-sphere S . It is a classical result of Brouwer and Kerekjarto from 1919 that such a finitegroup action on a 2-manifold is locally linear : each fixed point of a nontrivial elementhas a regular neighbourhood which is a 2-disk on which the cyclic subgroup fixing thepoint acts as a standard orthogonal rotation on the 2-disk. This implies easily that thequotient space S /G (the space of orbits) is a again a 2-manifold, or better a of some signature ( g ; n , . . . , n r ): an orientable surface of some genus g , with r branchpoints of orders n , . . . , n r which are the projections of the fixed points of the nontrivialcyclic subgroup of G (of orders n i > S → S /G is a branched covering . We choose some triangulation of thequotient orbifold S/G such that the branch points are vertices of the triangulation, andlift this triangulation to a triangulation of S ; then the projection S → S /G becomesa simplicial map. If the projection p is a covering in the usual sense (unbranched, i.e.without branch points), then clearly the Euler characteristic χ behaves multiplicatively,i.e. 2 = χ ( S ) = | G | χ ( S /G ) = | G | (2 − g ) (just multiplying Euler characteristics withthe order | G | of G ). In the case with branch points, we can correct this by subtracting | G | for each branch point and then adding | G | /n i (the actual number of points of S projecting to the i’th branch point), obtaining in this way the classical formula ofRiemann-Hurwitz : 2 = χ ( S ) = | G | (2 − g − r X i =1 (1 − n i )) . It is easy to see that the only solutions with positive integers of this equation are thefollowing (first two columns):(0; n, n ) , | G | = n ; G ∼ = Z n cyclic;(0; 2 , , n ) , | G | = 2 n ; G ∼ = D n dihedral;(0; 2 , , , | G | = 12; G ∼ = A tetrahedral;(0; 2 , , , | G | = 24; G ∼ = S octahedral;(0; 2 , , , | G | = 60; G ∼ = A dodecahedral.We still have to identify the groups G . For this, we first determine the finite subgroupsof the orthogonal group SO(3) and suppose that the action of G on S is orthogonal. Ofcourse the possibilities for the signatures of the quotient orbifolds S /G and the orders | G | remain the same, and for orthogonal actions one can identify the groups now as theorientation-preserving symmetry groups of the platonic solids. As an example, if thesignature is (0; 2,3,5) and | G | = 60, one considers a fixed point P of a cyclic subgroup Z of G and the five fixed points of subgroups Z closest to P on S ; these are thevertices of a regular pentagon on S which is one of the twelve pentagons of a regulardodecahedron projected to S , invariant under the action of G . Hence G , of order 60,3oincides with the orientation-preserving isometry group of the regular dodecahedron,the dodecahedral group A ; see [W, section 2.6] for more details. In this way one showsthat the finite subgroups of SO(3) are, up to conjugation, exactly the polyhedral groups as indicated in the list above: cyclic Z n , dihedral D n , tetrahedral A , octahedral S and dodecahedral A .As a consequence, returning to topological actions of finite groups G on S , the topolog-ical orbifolds S /G in the above list are geometric since they are homeomorphic exactlyto the quotients of S by the polyhedral groups. Hence the topological orbifolds S /G have a spherical orbifold structure (a Riemannian metric of constant curvature one, withsingular cone points of angles 2 π/n i ); lifting this spherical structure to S realizes S asa spherical manifold (i.e., with a Riemannian metric of constant curvature 1, withoutsingular points). Since the spherical metric on S is unique up to isometry this givesthe standard Riemannian S , and G acts by isometries now. Hence every topologicalaction of a finite group G on S is conjugate to an orthogonal action, that is finitegroup actions on S are linear (this can be considered as the orbifold geometrization indimension two , in the spherical case).Concluding and summarizing, finite group actions on S are locally linear , and thenalso linear ; the finite groups occurring are exactly the polyhedral groups, and everytopological action of such a group is geometric (or linear, or orthogonal), i.e. conjugateto a linear action.
3. Finite groups acting on S and finite subgroups of SO(4) .3.1. Geometrization of finite group actions on S We consider actions of a finite group G on the 3-sphere S now. The first question is ifsuch a topological action is locally linear ; by the normal form for orthogonal matrices, indimension three this means that an element with fixed points acts locally as a standardrotation around some axis (the orientation-preserving case). Unfortunately this is nolonger true in dimension three; after a first example of Bing from 1952 in the orientation-reversing case, Montgomery-Zippen 1954 gave examples also of orientation-preservingcyclic group actions on S with ”wildly embedded fixed point sets”, i.e. with fixedpoints sets which are locally not homeomorphic to the standard embedding of S in S .Obviously such actions are not locally linear, and in particular cannot be conjugate tosmooth or linear actions.We will avoid these wild phenomena in the following by concentrating on smooth or locally linear actions. Suppose now that a cyclic group G ∼ = Z p , for a prime p , actslocally linear on S with nonempty fixed point set; then it acts locally as a standardrotation around an axis, and by compactness this axis closes globally to an embeddedknot K ∼ = S in S . We note that also for a topological action of G , by general Smithfixed point theory the fixed point set of G is a knot K , i.e. an embedded S in S . If the4ction is locally linear, K is a tame knot as considered in classical knot theory (smoothor polygonal embeddings), otherwise K is a wild knot leading to the different field ofwild or Bing topology (see [Ro, chapter 3.I] for such wild phenomena in dimensionthree).So we will consider only locally linear actions in the following. Globally, the questionarises then which knots K can occur as the fixed point set of an action of a cyclic groupon S ; it is easy to see that the action of G ∼ = Z p is linear (conjugate to a linear action)if and only if K is a trivial knot (unknotted, i.e. bounding a disk in S ). The classical Smith conjecture states that K is always a trivial knot, and that consequently locallylinear actions of cyclic groups are linear. A positive solution of the Smith conjecture wasthe first major success of Thurston’s geometrization program for 3-manifolds (see [MB]).This has been widely generalized by Thurston then who showed that finite nonfree groupactions on closed 3-manifolds are build from geometric actions ( orbifold geometrizationin dimension three ), and recently by Perelman also for free actions of finite groups( manifold geometrization in dimension three ). As a consequence, every finite groupacting smoothly or locally linearly on S is geometric, i.e. conjugate to an orthogonalor linear action.Concluding, finite group actions on S are not locally linear, in general, but smoothor locally linear actions are linear; in particular, the finite groups acting smoothly orlocally linearly on S are exactly the finite subgroups of the orthogonal group SO(4).In the remaining part of this section we discuss the finite subgroups of the orthogonalgroup SO(4), starting with the relation between SO(3) and the unit quaternions. SO(3) and the unit quaternions S . The orthogonal group SO(3) is homeomorphic to the real projective space RP of di-mension three. In fact, by the normal form for orthogonal 3 × R around some oriented axis or diam-eter; parametrising the diameter by an rotation angle from − π to π , one obtains SO(3)by identifying diametral points on the boundary S of the 3-ball (since − π and π givethe same rotation), and consequently SO(3) is homeomorphic to RP .Hence the universal covering of SO(3) ∼ = RP ∼ = S / h± id i is the 3-sphere S . Con-sidering S as the unit quaternions, an orthogonal action of S on the 2-sphere S isobtained as follows. The unit quaternions S act on itself by conjugation x → q − xq ,for a fixed q ∈ S ; this action is clearly linear and also orthogonal. Since q fixes bothpoles 1 and -1 in S , it restricts to an orthogonal action on the corresponding equatorial2-sphere S in S , so this defines an element of the orthogonal group SO(3) and a grouphomomorphism S → SO(3) of Lie groups of the same dimension, with kernel ±
1; bystandard facts about Lie groups, this is the universal covering of SO(3) ∼ = S / h± i .5he finite subgroups of SO(3) are exactly the polyhedral groups Z n , D n , A , S and A .Their preimages in the unit quaternions S are the binary polyhedral groups Z n , D ∗ n , A ∗ , S ∗ , A ∗ (cyclic, binary dihedral, binary tetrahedral, binary octahedral or binary dodecahedral).Since S has a unique nontrivial involution -1, together with the cyclic groups of oddorder these are exactly the finite subgroups of the unit quaternions S , up to conjuga-tion. By right multiplication, they act freely and orthogonally on the 3-sphere S , andthe quotient spaces are examples of spherical 3-manifolds; for example, S / A ∗ is the Poincar´e homology 3-sphere .We note that the Lie group S has various descriptions: it occurs as the unitary groupSU(2) over the complex numbers, as the universal covering group Spin(3) of SO(3) overthe reals, and finally as the symplectic group Sp(1) over the quaternions (in fact, theunit quaternions). SO(4) as a central product S × Z S . Passing to the orthogonal group SO(4) now acting on the unit 3-sphere S ⊂ R , thereis an orthogonal action of S × S on S given by x → q − xq , for a fixed pair ofunit quaternions ( q , q ) ∈ S × S . This defines again a homomorphism of Lie groups S × S → SO(4) of the same dimension, with kernel Z generated by ( − , − S × S , and SO(4) is isomorphic tothe central product S × Z S of two copies of the unit quaternions (the direct productwith identified centers, noting that the center of the unit quaternions is isomorphic to Z generated by − S × Z S , the finite subgroups of SO(4) are, up to conjugation,exactly the finite subgroups of the central products P ∗ × Z P ∗ ⊂ S × Z S of two binary polyhedral groups P ∗ and P ∗ . The most interesting example of such agroup is the central product A ∗ × Z A ∗ of two binary dodecahedral groups which occursas the orientation-preserving symmetry group of the regular 4-dimensional 120-cell (afundamental domain for the universal covering group A ∗ of the Poincar´e homologysphere S / A ∗ is a regular spherical dodecahedron, and 120 copies of this dodecahedrongive a regular spherical tesselation of the 3-sphere S ; the vertices of this tesselationare the vertices of the regular euclidean 120-cell in R whose faces are 120 regulardodecahedra.)Concluding, the finite subgroups of SO(4) are exactly the subgroups of the centralproducts P ∗ × P ∗ of two binary polyhedral groups P ∗ and P ∗ . It is then an algebraic6xercise to classify the possible groups, up to isomorphism and up to conjugation (see[DV] for a list of the finite subgroups of SO(4), and also of O(4)).
4. Finite groups acting on S and finite subgroups of SO(5) . As we have seen in the previous sections, finite group actions on the 2-sphere are locallylinear, and then also linear. In dimension three, finite group actions are not locallylinear, in general, but by deep results of Thurston and Perelman, smooth or locallylinear actions on the 3-sphere are linear. In dimension four, also smooth or locallylinear actions are no longer linear, in general. In fact it has been shown by Giffenin 1966 that the Smith conjecture fails in dimension four, by constructing examples ofsmooth actions of a finite cyclic group on S whose fixed point sets are knotted 2-spheresin S (see [R, chapter 11.C]); in particular, such an action cannot be linear.Restricting again to smooth or locally linear actions, we consider now the problem statedin the introduction: which finite groups G admit a smooth orientation-preserving actionon the 4-sphere S ; also, what are the finite subgroups of SO(5)?Suppose that G is a finite group with an orientation-preserving, faithful, linear action on S ⊂ R ; using the language of group representations, this means that G has a faithful,orientation-preserving, real representation in dimension five. If such a representationis reducible (a direct sum of lower-dimensional representations), G is an orientation-preserving subgroup of a product of orthogonal groups O(3) × O(2) or O(4) × O(1), soone can reduce to lower dimensions.Suppose that the representation is irreducible but imprimitive ; this means that thereis a decomposition of R into proper linear subspaces which are permuted transitivelyby the group. Since the dimension five is prime, these linear subspaces have to be 1-dimensional (such a representation is then called monomial). The group of orthogonalmaps permuting the five factors R of R is the Weyl-group W = ( Z ) ⋊ S of inversionsand permutations of coordinates, i.e. the semidirect product of the normal subgroup( Z ) generated by the inversions and the symmetric group S of permutations of thefactors. Hence G is a subgroup of the Weyl-group W = ( Z ) ⋊ S in this case.There remains the case of an irreducible, primitive representation; this is the maincase which has been considered by various authors and for arbitrary dimension; a majorproblem here is to find the simple groups which admit such a representation (i.e., groupswithout a nontrivial proper normal subgroup). This leads into classical representationtheory of finite groups, and we will not go further into it. In fact, we gave the abovedescription mainly as a motivation for the next result on smooth or locally linear actionsof finite groups on the 4-sphere. 7 heorem 1. ([MeZ1]) A finite group G with a smooth or locally linear, orientation-preserving action on the 4-sphere, or on any homology 4-sphere, is isomorphic to one ofthe following groups:i) an orientation-preserving subgroup of O(3) × O(2) or O(4) × O(1) ;ii) an orientation-preserving subgroup of the Weyl group W = ( Z ) ⋊ S ;iii) A , S , A or S ;i’) if G is nonsolvable, a 2-fold extension of a subgroup of SO(4) . Note that the different cases of the Theorem are not mutually exclusive. The onlyindetermination remains case i’); in fact, in this case G should be isomorphic to asubgroup of O(4) and hence to an orientation-preserving subgroup of the group O(4) × O(1) of case i); however the proof in this case is not completed at present since manydifferent cases have to be considered, according to the long list of finite subgroups ofSO(4) (see [DV]).
Corollary 1.
A finite group G which admits an orientation-preserving action on ahomology 4-sphere is isomorphic to a subgroup of SO(5) or, if G is solvable, to a 2-foldextension of a subgroup of SO(4) . The symmetric group S acts orthogonally on R by permutation of coordinates, andalso on its subspace R defined by setting the sum of the coordinates equal to zero(this is called the standard representation of S ), and hence on the unit sphere S ⊂ R . Composing the orientation-reversing elements by -id, one obtains an orientation-presering action of S on S (alternatively, S acts on the 5-simplex by permuting itssix vertices, and hence on its boundary which is the 4-sphere.)For linear action, Theorem 1 and its proof easily give the following characterization ofthe finite subgroups of SO(5). Corollary 2.
Let G be a finite subgroup of the orthogonal group SO(5) . Then one ofthe following cases occurs:i) G is conjugate to an orientation-preserving subgroup of O(4) × O(1) or O(3) × O(2) (the reducible case);ii) G is conjugate to a subgroup of the Weyl group W = ( Z ) ⋊ S (the irreducible,imprimitive case);iii) G is isomorphic to A , S , A or S (the irreducible, primitive case). See the character tables in [C] or [FH] for the irreducible representations of the groupsin iii) (e.g. A occurs as an irreducible subgroup of all three orthogonal groups SO(3),SO(4) and SO(5)).It should be noted that the proof of Corollary 2 is considerably easier than the proof ofTheorem 1. For both Theorem 1 and Corollary 2 one has to determine the finite simple8roups which act on a homology 4-sphere resp. which admit an orthogonal action onthe 4-sphere. In the case of Theorem 1 this is based on [MeZ2, Theorem 1] whichemploys the Gorenstein-Harada classification of the finite simple groups of sectional2-rank at most four (see [Su2], [G1]). For the proof of Corollary 2 instead, this heavymachinery from the classification of the finite simple groups can be replaced by muchshorter arguments from the representation theory of finite groups.For solvable groups G instead, the proof of Theorem 1 is easier; here one can consider theFitting subgroup of G , the maximal normal nilpotent subgroup, which is nontrivial forsolvable groups. As a nilpotent group, the Fitting subgroup is the direct product of itsSylow p -subgroups, has nontrivial center and hence nontrivial cyclic normal subgroupsof prime order. A starting point of the proof of Theorem 1 is then the following lemmawhich shows some of the basic ideas involved. Lemma 1.
Let G be a finite group with a smooth, orientation-preserving action on ahomology 4-sphere. Suppose that G has a cyclic normal group Z p of prime order p ; bySmith fixed point theory, the fixed point set of Z p is either a 0-sphere S or a 2-sphere S (i.e., a mod p homology sphere of even codimension).i) If the fixed point set of Z p is a 0-sphere then G contains of index at most two asubgroup isomorphic to a subgroup of SO(4) . Moreover if G acts orthogonally on S then G is conjugate to a subgroup of O(4) × O(1) .ii) If the fixed point set of Z p is a 2-sphere then G is isomorphic to a subgroup of O(3) × O(2) . Moreover if G acts orthogonally on S then G is conjugate to a subgroupof O(3) × O(2) . Proof. i) Since Z p is normal in G , the group G leaves invariant the fixed point set S of Z p which consists of two points. A subgroup G of index at most two of G fixes bothpoints and acts orthogonally and orientation-preservingly on a 3-sphere, the boundaryof a G -invariant regular neighborhood of one of the two fixed points.If the action of G is an orthogonal action on the 4-sphere then G acts orthogonally onthe equatorial 3-sphere of the 0-sphere S and hence is a subgroup of O(4) × O(1), upto conjugation.ii) The group G leaves invariant the fixed point set S of Z p . A G -invariant regularneighbourhood of S is diffeomorphic to the product of S with a 2-disk, so G acts onits boundary S × S (preserving its fibration by circles). Now, by the geometrization offinite group actions in dimension three, it is well-known that every finite group action on S × S preserves the product structure and is standard, i.e. is conjugate to a subgroupof its isometry group O(3) × O(2).If G acts orthogonally on S then the group G leaves invariant S , the corresponding3-dimensional subspace in R as well as its orthogonal complement, so up to conjugationit is a subgroup of O(3) × O(2). 9 . Higher dimensions
Relevant in the context of linear actions on spheres is the classical
Jordan number :for each dimension n there is an integer j ( n ) such that each finite subgroup of thecomplex linear group GL n ( C ), and hence in particular also of its subgroup SO( n ), hasa normal abelian subgroup of index at most j ( n ) (we note that a lower bound for j ( n )is ( n + 1)! since the symmetric group S n +1 is a subgroup of GL n ( C ); see the commentsto [KP, Theorem 5.1] for an upper bound.) Whereas this is insignificant for abeliangroup, it implies that the order of a nonabelian simple groups acting linearly on S n is bounded by j ( n + 1); in particular, up to isomorphism there are only finitely manyfinite simple groups (always understood to be nonabelian in the following) which admit afaithful, linear action on S n (or equivalently, have a faithful, real, linear representationin dimension n + 1). For smooth or locally linear actions of finite simple groups onspheres and homology spheres, there is the following analogue. Theorem 2. ([GZ])
For each dimension n , up to isomorphism there are only finitelymany finite simple groups which admit a smooth or locally linear action on the n -sphere,or on some homology sphere of dimension n . We note that any finite simple group admits many smooth actions on high-dimensionalspheres which are not linear (conjugate to a linear action; see the survey [Da, section7]).It is natural to ask whether the Jordan number theorem can be generalized for all finitegroups acting on homology n -spheres. Since, as noted in the introduction, finite p -groupsadmitting a smooth action on some homology n -sphere admit also a linear action on S n ([DH]), it is easy to generalize the Jordan number theorem for nilpotent groups; sothe Jordan theorem remains true for the two extreme opposite cases of nilpotent groupsand simple groups, but at present we don’t know it for arbitrary finite groups.Not surprisingly, the proof of Theorem 2 requires the full classification of the finitesimple groups; we will present part of the proof in the following. We note that the proofof Theorem 2 permits to produce for each dimension n a finite list of finite simple groupswhich are the candidates for actions on homology n -spheres; then one can identify thosegroups from the list which admit a linear action on S n (or equivalently, have a faithful,real, linear representation in dimension n + 1), and try to eliminate the remaining onesby refined methods. For example, it is shown in [MeZ2-4] that the only finite simplegroup which admits an action on a homology 3-sphere is the alternating group A , andthat the only finite simple groups acting on a homology 4-sphere are the alternatinggroups A and A ; already these low-dimensional results require heavy machinery fromthe classification of the finite simple groups.Crucial for the proofs of Theorems 1 and 2 is a control over the minimal dimension ofan action of a linear fractional group PSL ( p ) and a linear group SL ( p ) (the latter is10he group of 2 × p elements,the former its factor group by the central subgroup Z = h± E i ); this is given by thefollowing: Proposition 1. ([GZ])
For a prime p ≥ , the minimal dimension of an action of alinear fractional group PSL ( p ) on a mod p homology sphere is ( p − / if p ≡ ,and p − if p ≡ , and these are also lower bounds for the dimension of such anaction of a linear group SL ( p ) . Whereas the groups PSL ( p ) admit linear actions on spheres of the corresponding di-mensions (see e.g. [FH]), for the groups SL ( p ) the minimal dimension of a linear actionon a sphere is p − p , that is strictly larger than the lower bounds given in Propo-sition 1, so the minimal dimension of an action on a homology sphere remains openhere.In analogy with Proposition 1, we need also the following result from Smith fixed pointtheory for elementary abelian p -groups ([Sm]). Proposition 2.
The minimal dimension of a faithful, orientation-preserving action ofan elementary abelian p -group ( Z p ) k on a mod p homology sphere is k if p = 2 , and k − if p is an odd prime. Considering commutator subgroups, the proof of the following lemma is an easy exercise(a finite central extension of G is a finite group with a central subgroup whose factorgroup is isomorphic to G ; a group is perfect if it coincides with its commutator subgroupor, equivalently, the abelianized group is trivial). Lemma 2.
If a finite group G has a perfect subgroup H then any finite central extensionof G contains a perfect central extension of H . For the proof of Theorem 2, Lemma 2 will be applied mainly when H is a linear groupSL ( q ), for a prime power q = p k (that is, over the finite field with p k elements); we notethat, for q ≥ ( q ) is the group itself (see [H, chapter V.25]) (and the only nontrivial perfectcentral extension of PSL ( q ) is SL ( q )).On the basis of Propositions 1 and 2 and the classification of the finite simple groups,we indicate now the Proof of Theorem 2.
Fixing a dimension n , we have to exclude all but finitely manyfinite simple groups. By the classification of the finite simple groups, a finite simplegroup is one of 26 sporadic groups, or an alternating group, or a group of Lie type([Co], [G1]). We can neglect the sporadic groups and have to exclude all but finitelymany groups of the infinite series. Clearly, an alternating group A m contains elementary11belian subgroups ( Z ) k of rank k growing with the degree m , so Proposition 2 excludesall but finitely many alternating groups and we are left with the infinite series of groupsof Lie type.We consider first the projective linear groups PSL m ( q ), for a prime power q = p k .The group PSL ( q ) has subgroups PSL ( p ) and ( Z p ) k (the subgroup represented by alldiagonal matrices with entries one on the diagonal, isomorphic to the additive groupof the field with p k elements), and by Propositions 1 and 2 only finitely many primes p and prime powers p k can occur. If m ≥ m ( q ) has subgroups SL ( q )and SL ( p ); again Proposition 1 excludes all but finitely many primes p and, since alsoSL ( q ) has an elementary abelian subgroup ( Z p ) k , by Proposition 4 only finitely manypowers p k of a fixed prime p can occur. Concluding, only finitely many prime powers p k can occur for a fixed dimension n . Note that, in a similar way, also for the groupsSL ( q ) only finitely many values of q can occur. We still have to bound m ; if q is not apower of two, then PSL m ( q ) has a subgroup ( Z ) m − represented by diagonal matriceswith entries ± m is bounded by Proposition 2. If q is a power oftwo, one may consider instead subgroups ( Z p ) [ m/ < SL ( p ) [ m/ < PSL m ( q ) and againapply Proposition 2.This finishes the proof of Theorem 2 for the case of the projective linear groups PSL m ( q ).The proof for the unitary groups PSU m ( q ) and the symplectic groups PSp m ( q ) issimilar, noting that there are isomorphismsPSU ( q ) ∼ = PSp ( q ) ∼ = PSL ( q ) , SU ( q ) ∼ = Sp ( q ) ∼ = SL ( q );in particular, if m ≥ m ≥
4, the latter groups are subgroups of both PSU m ( q )and PSp m ( q ), so we can conclude as before.The last class of classical groups are the orthogonal groups Ω m +1 ( q ) = P Ω m +1 ( q ) and P Ω ± m ( q ) (the latter stands for two different groups which are simple if m ≥ ( q ) ∼ = PSL ( q ) , P Ω +4 ( q ) ∼ = PSL ( q ) × PSL ( q ) , P Ω − ( q ) ∼ = PSL ( q )(see [Su, p.384]). By canonical inclusions between orthogonal groups and the casesconsidered before, this leaves again only finitely many possibilities.Next we consider the exceptional groups G ( q ), F ( q ), E ( q ), E ( q ), E ( q ) as well asthe Steinberg triality groups D ( q ). By [St, Table 0A8], [GL, Table 4-1], up to centralextensions there are inclusions E ( q ) > F ( q ) > D ( q ) > G ( q ) > PSL ( q ) , E ( q ) > PSL ( q ) , E ( q ) > PSL ( q ) . Applying Lemma 1 we reduce to subgroups SL ( q ) in all cases.12inally, there remain the twisted groups E ( q ) , D ( q ) , Sz (2 m +1 ) = B (2 m +1 ) , G (3 m +1 ) , F (2 m +1 ) . By [St, Table 0A8], [GL, Table 4-1], up to central extensions there are inclusions E ( q ) > F ( q ) and D ( q ) > G ( q ) (already considered before); the Suzuki groups Sz (2 m +1 ) have subgroups ( Z ) m +1 ([G1, p.74]), the Ree groups G (3 m +1 ) subgroupsPSL (3 m +1 ) ([G2, p.164]) and the Ree groups F (2 m +1 ) subgroups SU (2 m +1 ) ([GL,Table 4-1]), so in all these cases some previously considered case applies.This completes the proof of Theorem 2. Acknowledgment.
I want to thank the colleagues and friends from the Universitiesof Novosibirsk, Kemerovo and Krasnojarsk for their great hospitality during my visit injune 2011 which gave origin to the present notes.
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