When Is a Group the Union of Proper Normal Subgroups?

Let H be a finite group having a fixed point free au tomorph ism c~ of order p". Consider the semidirect product G = (c~)H. It is well known that (eh) v" = 1 if h 9 H (see [3], p. 334). Put K = ( ev ) H. Then G # K and the elements in G K are p-elements. This last si tuation was considered by Kurzweil in [7]. It includes as a special case the groups having a proper generalized Hughes subgroup, i.e. those verifying G + Hr, (G) where Hp, (G) = ( x 9 G I xl" Je 1). A classical result of Hughes-Thompson and Kegel assures that if G :# H v (G) then H v (G) is ni lpotent (see [5] and [6]). Assuming that G is solvable Kurzweil showed that the Fi t t ing length of Hr, (G) (and hence that of G) is bounded by a function of n (see [7]). His bound for exceptional primes (in the Hal l -Higman sense) was improved by Har t ley and Rae as a product of their work in [4]. More recently Meixner obtained a l inear bound in [8]. Finally, in [2], the best possible bound f (Hr, (G)) < n was obtained for p odd. The case p = 2 is open. The purpose of this note is to consider the general problem. We may assume that G = ( x ) K, G K consists of p-elements and the order of x is, say, p". Assuming that G is solvable, what can be said about its Fi t t ing length? In [7] Kurzweil considered the case n = I and showed that f (K) < 2. Here we prove that f (K) < n + 1 if p is odd and the bound is best possible. The result is false for p = 2 even in the case n = 2. Our theorem is a new appl icat ion of the non-coprime Shult type theorems stated in [2]. There is another problem connected to this. Let G be a finite group having a proper subgroup H and a proper normal subgroup N of H such that H c~ H ~ < N if g 9 G H. Then G is said to be a Frobenius-Wie landt group (see [1] for more details and notation). We write (G, H, N) to indicate this situation. A theorem of Wielandt (see [1] for example) assures that, in such conditions, there exists a normal subgroup K of G such that G K = ~) (H -N) o, G = H K and H c~ K = N. Assume that H is a p-group. Then osG G K consists of p-elements. Thus we are in the above situation. Conversely, if G is p-solvable and K is a normal subgroup of G such that G K consists of p-elements then taking P 9 $1, (G) we have that (G, P, P c~ K) is an F W group. To show this observe that if x 9 G K then x acts f.p.f, on every x-invariant p '-section of K. Suppose that y 9 P c~ Po where g is a nontrivial p ' -element of G. As K is p-solvable we have a p '-section A/B of K where A and B are normal in G and g 9 A B. Then [y, g 1] 9 p c~ A < B. Thus y 9 P c~ K.

Call a group G hypersolvable if it has an ascending series with G/CG(A) solvable for each factor A of the series. In this article we establish some basic facts about hypersolvable groups. We also prove that if G is a perfect Fitting p-group such that every proper subgroup is contained in a proper normal subgroup, then G has a proper non-hypersolvable subgroup.

Necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This result is manifest in the case of random walks on finite groups by a statement about the support of the driving probability: a random walk on a finite group is ergodic if and only if the support is not concentrated on a proper subgroup, nor on a coset of a proper normal subgroup. The study of random walks on finite groups extends naturally to the study of random walks on finite quantum groups, where a state on the algebra of functions plays the role of the driving probability. Necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group are given on the support projection of the driving state.

Let G be a finite group and N a nontrivial proper normal subgroup of G. A.R. Camina introduced the class of finite groups G, which extends Frobenius groups, satisfying that for all g∈G−N and n∈N, gn is conjugate to g. He proved that under these assumptions one of three possibilities occurs: G is a Frobenius group with kernel N; or N is a p-group; or G/N is a p-group. In this paper we extend this class of groups by investigating the structure of those finite groups G having a nontrivial proper normal subgroup N such that gn is conjugate to either g or g−1 for all g∈G−N and all n∈N.

A finite group G is called a generalized Frobenius group with kernel F if F is a proper nontrivial normal subgroup of G, and for every element Fx of prime order p in the quotient group G/F, the coset Fx of G consists of p-elements. We study generalized Frobenius groups with an insoluble kernel F. It is proved that F has a unique non- Abelian composition factor, and that this factor is isomorphic to $$ {\mathrm{L}}_2\left({3}^{2^{\mathrm{l}}}\right) $$ for some natural number l. Moreover, we look at a (not necessarily finite) group generated by a coset of some subgroup consisting solely of elements of order three. It is shown that such a group contains a nilpotent normal subgroup of index three.

We recall the following definition (see [1]):A finite group G is said to be a Frobenius–Wielandt group provided that there exists a proper subgroup H of G and a proper normal subgroup N of H such that H∩Hg≦N if g∈G–H. Then H/N is said to be the complement of (G, H, N) (see [1] for more details and notation).

A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a union of n distinct conjugacy classes of G. In some research papers, the question of finding all positive integer n such that there is an n-decomposable finite group was posed.

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK G = {ncc(A)¦A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z6, D8, Q8, S4, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].

In this article the study of generalized Frattini subgroups of finite groups, developed by J. C

This note is concerned with the following problem. Let H denote a subgroup of a finite group G and let L denote a linear or one dimensional representation (i.e., a character) of H. We assume throughout that the field F is algebraically closed and is either of characteristic 0 or of prime characteristic which does not divide the order of any groups under consideration. Let GIL denote the corresponding induced representation of G. How many distinct (i.e., nonequivalent) irreducible representations appear in the decomposition of GI L into irreducible parts? (This number is just the central intertwining number of GI L, which is denoted by Ct(GI L). Cf. [1].) More specifically, we are interested in determining an upper bound on the number of distinct irreducible representations which will appear, purely in terms of the way H is embedded in G, and in terms which do not depend on the particular linear representation L of H. Two such bounds come quickly to mind. The number of classes (of conjugates) of the super group G, which we denote { G: e}, is clearly an upper bound. Dimension considerations also give [G: H] as an upper bound. We now introduce a new group theoretic invariant which heuristically is a measure of the manner in which the classes of G are distributed among the H-cosets of G. DEFINITION. Let H be a (not necessarily normal) subgroup of a finite group G. For each normal subset N of G, let +1(N) denote the number of classes (of conjugates) of G contained in N. Let +2(N) denote the number of right H-cosets of G which have nonzero intersection with N. Let +(N) = { G: e} -45(N) +42(N). We then define the embedding number of H in G, denoted by (G: H), to be the minimum of the +(N), as N is taken over all normal subsets of G. We remark that a definition of 42 using left cosets would yield the same value for (G: H) since N-' intersects the same number of left cosets as N does right cosets. Taking N= {e } where e is the identity element of the group we have (G: H) {G: e}. Taking N=G we have (G: H)? [G: H]. If H$ G, it is easy to verify that (G: H) > 1. If His a proper normal subgroup, then, taking N=H we have (G:H)<{G:e}. In the case where H is a normal subgroup of G, another number associated with the embedding of H in G is the number of classes in the factor group G/H. We call this the class number of H in G and denote it by { G: H}.

A finite group $$G$$ is said to be a generalized Frobenius group with kernel$$F$$, if $$F$$ is a proper nontrivial normal subgroup of $$G$$ and for every element $$Fx$$ of prime order of the quotient group $$G/F$$ the coset $$Fx$$ of the group $$G$$ over $$F$$ has only $$p$$-elements for some prime $$p$$ depending on $$x$$. This article considers generalized Frobenius groups with insoluble kernel. We prove that a quotient group of a generalized Frobenius group over its insoluble kernel is a 2-group.

A finite group is inseparable, it does not split over any proper nontrivial normal subgroup; that is, if it has no nontrivial semidirect product decompositions. This paper investigates two classes of finite inseparable p-groups and, for p ≥ 3, establishes a necessary and sufficient condition for inseparability.

In this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group R has a nontrivial proper normal subgroup N with the property that the automorphism group of the digraph fixes each N-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.

A scheme of construction of infinite groups, other than simple groups, free groups of infinite rank and the infinite cyclic group, which are isomorphic to all their non-trivial normal subgroups is presented. Some results about the automorphism groups of simple infinite groups are also obtained. In particular, it is proved that there is an infinite group G of any sufficiently large prime exponent p (or which is torsion-free) all of whose proper subgroups are cyclic, and such that the groups Aut G and Out G are isomorphic. The proofs use the technique of graded diagrams developed by A. Yu. Ol'shanskii. 1991 Mathematics Subject Classification: 20F05, 20F06.

Suppose G is a finite group and C(G) denotes the set of all conjugacy classes of G. The normal graph of G, N(G), is a finite simple graph such that V(N(G)) = C(G). Two conjugacy classes A and B in C(G) are adjacent if and only if there is a proper normal subgroup N such that A U B ? N. The aim of this paper is to study the normal graph of a finite group G. It is proved, among other things, that the groups with identical character table have isomorphic normal graphs and so this new graph associated to a group has good relationship by its group structure. The normal graphs of some classes of finite groups are also obtained and some open questions are posed.