Generalized Frattini Subgroups of Finite Groups. II

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.

The purpose of this paper is to generalize some of the fundamental properties of the Frattini subgroup of a finite group. For this purpose we call a proper normal subgroup H of G a generalized Frattini subgroup if and only if G = NG(P) for each normal subgroup L of G and each Sylow p-subgroup P, p is a prime, of L such that G = HNG(P). Here NG(P) is the normalizer of P in G. Among the generalized Frattini subgroups of a finite nonnilpotent group G are the center, the Frattini subgroup, and the intersection L(G) of all selfnormalizing maximal subgroups of G. The product of two generalized Frattini subgroups of a group G need not be a generalized Frattini subgroup, hence G may not have a unique maximal generalized Frattini subgroup. Let H be a generalized Frattini subgroup of G and let K be normal in G. If K/H is nilpotent, then K is nilpotent. Similarly, if the hypercommutator of K is contained in H, then K is nilpotent. We consider the Fitting subgroup FίG) of a nonnilpotent group G, and prove F(G) is a generalized Frattini subgroup of G if and only if every solvable normal subgroup of G is nilpotent. Now let H be a maximal generalized Frattini subgroup of a finite nonnilpotent group G. Following Bechtell we introduce the concept of an iϊ-series for G and prove that if G possesses an iJ-series, then H = L(G).

The general problem, with a particular instance of which the present paper is concerned, is to obtain a description of the local structure of a group from information about the global structure. The aspect of local structure investigated here is the embedding of subgroups, especially of nilpotent subgroups in finite soluble groups. A classification of embeddings of subgroups in finite groups by means of an arithmetic function called abnormal depth was proposed in [6]. Let H be a subgroup of a finite group G. Then a(G:H), the abnormal depth of H in G, is the least number of abnormal links appearing in any balanced chain of subgroups connecting H to G, that is a chain for which each link is either normal or abnormal. Thus a (G:H)= 0 if and only if H is subnormal in G; and a(G:P)__< 1 for every subgroup P of G of prime power order. It was shown in [6] that if H is a nilpotent subgroup of a finite soluble group G, of nilpotent length n, then a (G: H) =< n - 1. Here in w 1 we examine in greater detail the easiest non-trivial case, in which n = 2, and then in w 2 prove certain supplementary results for n = 3 and n = 4. Some simple wreath product properties are established in w 3 and used in w 4 for the construction of examples showing that the embedding results obtained cannot be improved in various obvious ways. Notation and terminology follow common usage. If t; and ~ are classes of groups, then 3s ~ denotes the class of all groups G having a normal subgroup X such that X e 3~ and G/X e ~. This defines a composition of classes of groups which in general is not associative. However, we shall deal only with classes of which the composition is associatNe, and we may therefore omit brackets from products of more than two classes. Since we shall be concerned exclusively with finite groups, we take 91 to denote the class of finite nilpotent groups and 9.1 the class of finite abelian groups. Then for any positive integer n, 9l" is the class of finite soluble groups of nilpotent lengths <__ n; and 9.I" is the class of finite soluble groups of derived lengths __< n. Henceforth the term group is understood to mean finite group. Then any group G has a unique smallest normal subgroup L such that G/L is nilpotent: G/L is called the 91-residual ofG. IfH is any subgroup of G, then there is a unique smallest normal subgroup of G containing H, called the normal closure of H in G and denoted by Ha; and a unique smallest subnormal subgroup of G containing H, called the subnormal closure of H in G and (following Wielandt [8]) denoted by H'" a. If H a = G, we shall say that H is contranormal in G. Then, for any subgroup H of G, it is clear that H is contranormal in H'" a. (This is to be compared with the fact that the hypernormalizer NE(H ) of H in G is self-normalizing in G.) An abnormal subgroup is both self-normalizing and

(2002). When Is a Group the Union of Proper Normal Subgroups? The American Mathematical Monthly: Vol. 109, No. 5, pp. 471-473.

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}.

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.

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

A subgroup H of a finite group G is called a c*-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ⊂ K is an S-quasinormal embedded subgroup of G. In this paper, the structure of a finite group G with some c*-normal maximal subgroups of Sylow subgroups is characterized and some known related results are generalized.

According to P. Hall, a subgroup \(H\) of a finite group \(G\) is called pronormal in \(G\) if, for any element \(g\) of \(G\), the subgroups \(H\) and \(H^{g}\) are conjugate in \(\langle H,H^{g}\rangle\). The simplest examples of pronormal subgroups of finite groups are normal subgroups, maximal subgroups, and Sylow subgroups. Pronormal subgroups of finite groups were studied by a number of authors. For example, Legovini (1981) studied finite groups in which every subgroup is subnormal or pronormal. Later, Li and Zhang (2013) described the structure of a finite group \(G\) in which, for a second maximal subgroup \(H\), its index in \(\langle H,H^{g}\rangle\) does not contain squares for any \(g\) from \(G\). A number of papers by Kondrat’ev, Maslova, Revin, and Vdovin (2012–2019) are devoted to studying the pronormality of subgroups in a finite simple nonabelian group and, in particular, the existence of a nonpronormal subgroup of odd index in a finite simple nonabelian group. In The Kourovka Notebook, the author formulated Question 19.109 on the equivalence in a finite simple nonabelian group of the condition of pronormality of its second maximal subgroups and the condition of Hallness of its maximal subgroups. Tyutyanov gave a counterexample \(L_{2}(2^{11})\) to this question. In the present paper, we provide necessary and sufficient conditions for the pronormality of second maximal subgroups in the group \(L_{2}(q)\). In addition, for \(q\leq 11\), we find the finite almost simple groups with socle \(L_{2}(q)\) in which all second maximal subgroups are pronormal.

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].

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.

A subgroup H is called c -normal in a group G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H G , where H G ≕ Core( H ) is the maximal normal subgroup of G which is contained in H . We obtain the c -normal subgroups in symmetric and dihedral groups. Also we find the number of c -normal subgroups of order 2 in symmetric groups. We conclude by giving a program in GAP for finding c -normal subgroups. AMS Classification : 20D25. Keywords : c -normal, symmetric, dihedral. Introduction The relationship between the properties of maximal subgroups of a finite group G and the structure of G has been studied extensively. The normality of subgroups in a finite group plays an important role in the study of finite groups. It is well known that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G . In Wang introduced the concept of c -normality of a finite group. He used the c -normality of a maximal subgroup to give some conditions for the solvability and supersolvability of a finite group. For example, he showed that G is solvable if and only if M is c -normal in G for every maximal subgroup M of G . In this paper, we obtain the c -normal subgroups in symmetric and dihedral groups, and also we find the number of c -normal subgroups of order 2 in symmetric groups.

g*: For each normal subgroup N$ 'P(G), each reduced product of G over N is a semidirect product. (G = NB is a reduced product over a normal subgroup N by a subgroup B iff B does not contain a proper subgroup B* such that G = NB*.) F. Gross [5] has shown that for a finite solvable group G having 4!(G) = 1, splitting over each normal subgroup is sufficient for the subgroup lattice to be comple

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.