Finite groupG Research Articles

Q-admissibility of certain nonsolvable groups

A finite groupG is calledQ-admissible if there exists a finite dimensional central division algebra overQ, containing a maximal subfield which is a Galois extension ofQ with Galois group isomorphic toG. It is proved thatS5−, one of the two nontrivial central extensions ofS5 byZ/2Z, isQ-admissible. As a consequence of that result and previous results of Sonn and Stern, every finite Sylow-metacyclic group, havingA5 as a composition factor, isQ-admissible.

The fitting length of solvableH pn-groups

For a (finite) groupG and some prime powerpn, theHpn-subgroupHpn (G) is defined byHpn (G)=〈xeG|xpn≠1〉. A groupH≠1 is called aHpn-group, if there is a finite groupG such thatH is isomorphic toHpn (G) andHpn (G)≠G. It is known that the Fitting length of a solvableHpn-group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableHpn-groups having Fitting lengthn, this result is “almost” best possible.

Dimensions of irreducible modules over twisted group algebras

LetN be a normal subgroup of a finite groupG, letF be an algebraically closed field, letα∈Z 2(G, F *) and letV be an irreducible module over the twisted group algebraF α. If charF=p>0 divides (G∶N), assume thatG/N isp-solvable. It is proved that dim F V divides (G∶N)d whered is the dimension of an irreducible constituent ofV N. The special case whereα=1 andN is abelian yields a well-known theorem of Dade [3]. Another special case, namely whereN is abelian, charF∤(G∶N) and the restriction ofα ofNxN is a coboundary is a generalization of the main result of Ng [5].

More on the Schur group of a commutative ring

The Schur group of a commutative ring, R, with identity consists of all classes in the Brauer group of R which contain a homomorphic image of a group ring RG for some finite group G. It is the purpose of this article to continue an investigation of this group which was introduced in earler work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well‐known Schur index in the global or local field case.

More on the Schur group of a commutative ring

The Schur group of a commutative ring, R, with identity consists of all classes in the Brauer group of R which contain a homomorphic image of a group ring RG for some finite group G. It is the purpose of this article to continue an investigation of this group which was introduced in earlier work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well‐known Schur index in the global or local field case.

Conjugacy classes of maximal nilpotent subgroups

It is shown that in an arbitrary finite groupG, any two maximal nilpotent subgroups ofG whose intersection contains its own centralizer inG, are conjugate inG.

The computations of some Schur indices

Let χ be an irreducible character of a finite groupG. Letp=∞ or a prime. Letm p (χ) denote the Schur index of χ overQ p , the completion ofQ atp. It is shown that ifx is ap′-element ofG such that $$X_u \left( x \right) \in Q_p \left( X \right)$$ for all irreducible charactersX u ofG thenm p (χ)/vbχ(x). This result provides an effective tool in computing Schur indices of characters ofG from a knowledge of the character table ofG. For instance, one can read off Benard’s Theorem which states that every irreducible character of the Weyl groupsW(E n), n=6,7,8 is afforded by a rational representation. Several other applications are given including a complete list of all local Schur indices of all irreducible characters of all sporadic simple groups and their covering groups (there is still an open question concerning one character of the double cover of Suz).

Nilpotent injectors in symmetric groups

AnN-Injector in an arbitrary finite groupG is defined as a maximal nilpotent subgroup ofG, containing a subgroupA ofG of maximal order satisfying class(A)>=2. Among other results theN-Injectors of Sym(n) are determined and shown to consist of a unique conjugacy class of subgroups of Sym(n).

Power automorphisms of finitep-groups

For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp2 possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.

Sylow-metacyclic groups andQ-admissibility

A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2a 3b and to a list of known “almost simple” groups.

Prime ideals in crossed products of finite groups

LetR * G be a crossed product of the finite groupG over the ringR. In this paper we discuss the relationship between the prime ideals ofR*G and theG-prime ideals ofR. In particular, we show that Incomparability and Going Down hold in this situation. In the course of the proof, we actually completely describe all the prime idealsP ofR*G such thatP∩R is a fixedG-prime ideal ofR. As an application, we prove that ifG is a finite group of automorphisms ofR, then the prime (primitive) ranks ofR and of the fixed ringRG are equal provided •G•−∈R. In an appendix, we extend some of these 3 results to crossed products of the infinite cyclic group.

Augmentation quotients of group rings and symmetric powers

LetGbe a group with the lower central seriesLetwhere Σ runs over all non-negative integersa1,a2,…,ansuch thatandis theaith symmetric power of the abelian groupGi/Gi+1whereLetI(G) be the augmentation ideal ofGin, the group ring ofGover. Define the additive groupQn(G) =In(G) /In+1(G) for anyn≥ 1. Then it is well known thatQ1(G) ≅W1(G) for any groupG. Losey (4,5) proved thatQ2(G) ≅W2(G) for any finitely generated groupG. Furthermore recently Tahara(12) proved thatQ3(G) is a certain precisely defined quotient ofW3(G) for any finite groupG.

On automorphism groups of Cayley graphs

LetXG,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(XG,H) must contain the regular subgroupLG corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(XG,H)∶LG] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.

Goldschmidt’s 2-signalizer functor theorem

A solvableA-signalizer functorϑ assigns to any non-identity elementx of the abelian 2-subgroupA of the finite groupG anA-invariant solvable 2′-subgroupθ(CG(x)) ofCG(x) such thatθ(CG(x)) ∩CG(y) ⊆ϑ(CG(y)) for allx, y ∈ A#.θ is called complete ifG has a solvableA-invariant 2′-subgroupK=θ(G) such thatCk(x)=θ(CG(x)) for everyx ∈ A#.

A stability theorem for minimum edge graphs with given abstract automorphism group

Given a finite abstract group G \mathcal {G} , whenever n n is sufficiently large there exist graphs with n n vertices and automorphism group isomorphic to G \mathcal {G} . Let ( G , n ) (\mathcal {G},n) denote the minimum number of edges possible in such a graph. It is shown that for each G \mathcal {G} there always exists a graph M M such that for n n sufficiently large, e ( G , n ) e(\mathcal {G},n) is attained by adding to M M a standard maximal component asymmetric forest. A characterization of the graph M M is given, a formula for e ( G , n ) e(\mathcal {G},n) is obtained (for large n n ), and the minimum edge problem is re-examined in the light of these results.

On 2-sylow intersections

Letz be an involution in the finite groupG and suppose thatz belongs to the center of a Sylow subgroup ofG. Ifz belongs to a unique Sylow subgroup ofG and ifG is not a trivial intersection group, thenG is not a simple group.

On the number of automorphisms of a finite group

A new bound is obtained for the function g ( h ), whose existence was proved by Ledermann & Neumann (1956), such that p h divides the order of the automorphism group of a finite group G , if p g ( h ) divides the order of G .