On finite groups in which some maximal invariant subgroups have indices a prime or the square of a prime

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.

If G is a finite group, we say that a series of subgroups G = Go > G > * > Gn= 1 is a maximal series of length n of G if Gi is a maximal subgroup of Gi1, 1 _ i < n. A subgroup H of G is called mth maximal in G if there exists at least one maximal series H= Gm< Gm.,-< <Go= G. Groups all of whose second, third and fourth maximal subgroups are invariant have been completely classified, see Janko [4], where the relevant results are enumerated. Further, those finite simple groups whose fifth maximal subgroups are trivial have been found by Janko [5]. Since Janko [6] has announced the discovery of a new simple group J, whose sixth maximal subgroups are all trivial, it may be of interest to classify those finite simple groups whose maximal chains are of length at most six. Thompson [8] has given the following DEFINITION. We say that a finite group G is an N-group if the normalizer of any nontrivial solvable subgroup is itself solvable. In the Main Theorem of [8], all simple N-groups are classified. These are the following groups: PSL(2, q), q a prime power greater than 3, PSL(3, 3), M,,, A7, Sz(22n+1) and PSU(3, 32). Since these groups have been studied elsewhere in great detail, and have been variously characterized group theoretically, we will consider these groups as known. Then we have the

The paper contains two theorems generalizing the theorems of Huppert concerning the characterization of supersolvable and p-supersolvable groups, respectively. The first of these gives a new approach to prove Huppert's first named result. The second one has numerous applications in the paper. The notion of balanced pairs is introduced for non-conjugate maximal subgroups of a finite group. By means of them some new deep results are proved that ensure supersolvability of a finite group.

A subgroup of a finite group G is said to be second maximal if it is maximal in every maximal subgroup of G that contains it. A question which has received considerable attention asks: can every positive integer occur as the number of the maximal subgroups that contain a given second maximal subgroup in some finite group G? Various reduction arguments are available except when G is almost simple. Following the classification of the finite simple groups, finite almost simple groups fall into three categories: alternating and symmetric groups, almost simple groups of Lie type, sporadic groups and automorphism groups of sporadic groups. This thesis investigates the finite alternating and symmetric groups, and finds that in such groups, except three well known examples, no second maximal subgroup can be contained in more than 3 maximal subgroups.

For G a group and M a subgroup of G, we say that a subgroup A of G is a supplement to M in G, if G = MA. We prove the conjecture of O.H. Kegel that a finite group whose maximal subgroups admit an abelian supplement is soluble. But this condition does not characterize the soluble groups among the finite groups. We prove that a finite group G is soluble if and only if every maximal subgroup M of G admits a supplement whose commutator subgroup is contained in M. Moreover, we determine the finite groups whose maximal subgroups have a nilpotent (resp. soluble) supplement. The latter groups still deserve a further analysis.

Let $\sigma~=\{\sigma_{i}\,|\,~i\in~I\}$ be some partition of the set of all primes $\Bbb{P}$.A set ${\cal~H}$ of subgroups of $G$ is said to be a complete Hall $\sigma~$-set of $G$ if every member $\ne~1$ of ${\cal~H}$ is a Hall $\sigma~_{i}$-subgroup of $G$, for some $i\in~I$, and $\cal~H$ contains exactly one Hall $\sigma~_{i}$-subgroup of $G$ for every $\sigma~_{i}\in~\sigma~(G)$. A subgroup $H$ of $G$ is said to be: $\sigma$-permutable or $\sigma$-quasinormal in $G$ if $G$ possesses a complete Hall $\sigma$-set ${\cal~H}$ such that $HA^{x}=A^{x}H$ for all $A\in~{\cal~H}$ and $x\in~G$: ${\sigma}$-subnormal in $G$ if there is a subgroup chain $A=A_{0}\leq~A_{1}\leq~\cdots~\leq~A_{t}=G$ such that either $A_{i-1}\trianglelefteq~A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a finite $\sigma_{i}$-group for some $\sigma_{i}\in~\sigma$ for all $i=1,~\ldots,~t$. If $M_n~ $n$-maximal subgroup of $G$. If each $n$-maximal subgroup of $G$ is $\sigma$-subnormal ($\sigma$-quasinormal, respectively) in $G$ but, in the case $~n~>~1$, some $(n-1)$-maximal subgroup is not $\sigma$-subnormal (not $\sigma$-quasinormal, respectively) in $G$, we write $m_{\sigma}(G)=n$ ($m_{\sigma~q}(G)=n$, respectively). In this paper, we show that the parameters $m_{\sigma}(G)$ and $m_{\sigma~q}(G)$ make possible to bound the $\sigma$-nilpotent length $l_{\sigma}(G)$ (see below the definitions of the terms employed), the rank $r(G)$ and the number $|\pi~(G)|$ of all distinct primes dividing the order $|G|$ of a finite soluble group $G$. We also give the conditions under which a finite group is $\sigma$-soluble or $\sigma$-nilpotent, and describe the structure of a finite soluble group $G$ in the case when $m_{\sigma}(G)=|\pi~(G)|$. Some known results are generalized.

Generation and random generation: From simple groups to maximal subgroups

Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of H , where H is the normal closure of H in G, that is, the smallest normal subgroup of G containing H. A group G is called an HNE 2-group if all cyclic subgroups of order 2 and 4 of G are Hall normally embedded in G. In this paper, we prove that all HNE 2-groups are 2-nilpotent. Furthermore, we also characterize the structure of finite group all of whose maximal subgroups are HNE 2-groups. Finally, we determine finite non-solvable groups all of whose second maximal subgroups are HNE 2-groups.

Clearly a group G is solvable if all of its chief factors are abelian, in fact G is solvable if each of its maximal subgroups avoids at least one abelian chief factor. In [4] it was announced that G is solvable if each of its maximal subgroups M avoids an abelian section derived from a maximal element of the index complex of M. Also announced was a characterization of the maximal normal solvable subgroup of a group as the intersection of certain maximal subgroups of the group. In this note proofs of these results are presented with some related results. Only finite groups are treated. I. Statements of definitions and results. Let M be a maximal subgroup of group G, M < G. A subgroup C of G is said to be a completion of M in G if C is not contained in M while every proper subgroup of C which is normal in G, is contained in M. The set, I(M), of all completions of M is called the index complex of M in G. The second restriction on a completion C insures that the product of all normal subgroups of G which are proper subgroups of C is itself a proper subgroup of C. It is convenient to define the strict core of a subgroup H + I of G to be the product of all normal subgroups of G which are proper subgroups of H; the strict core of H is denoted by k(H) = kG(H ). Clearly k(H) is proper in H when H~ G but k(H) can differ from H even when H is normal in G, for example, when H is a normal cyclic p-group. (So the strict core of a subgroup can differ from the core.) It follows then that subgroup C of group G is a completion of maximal subgroup M of G, i.e., C ~ I(M), provided (i) G = (C, M) and (ii) k(C) < M. Proposition. The index complex of a maximal subgroup M of group G is nonempty. In particular I (M) contains a normal subgroup of G. Clearly the collection of normal subgroups of G which do not lie in M is nonempty; chose C to be minimal in this partially ordered set. Then CM--(C,M) = G and k(C) < M, so that C ~ Z (M). [] If C is a normal completion of maximal subgroup M of group G, i.e., C< G and C ~ I(M), then C/k(C) is a chief factor of G which is avoided by M. M avoids a chief factor H/K of G if M contains K but not H, in which case H E I (M) and k (H) = K. The set I(M) is partially ordered by set inclusion; maximal elements of I(M) are called

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.

i, Introduction. In this paper we study groups with a locally nilpotent maximal subgroup. Our purpose is to find which conditions on the group G, or on the maximal locally nilpotent subgroup M, ensure the local solvability of G. tn the theory of finite groups there is a well-known result of Thompson, which says that if G is a finite group with a maximal subgroup M which is nilpotent and of odd order, then G is solvable (see [9]). The restriction on the order of M was due to the fact that, in order to obtain the above result, Thompson uses his famous J Theorem to prove the p-nilpotency of G/M~ (i,e. the existence of a normal p-complement) for the primes p involved in M / M G (M o is the core of M in G). Thus p must be # 2. Deskins and Janko proved that G is solvable even when the order of M is even, provided the Sylow 2-subgroup of M has class at most 2 (see [2] and [6]), The general concern with the Sylow 2-subgroup of M is due to the existence of a counterexample: namely PSL (2, I7), which is simple and contains a maximal nilpotent subgroup isomorphic to D16, hence a 2-subgroup of nilpotency class 3. In the case that G is an infinite periodic linear group contained in GL (n, K), Wehrfritz proved that if M is a maximal subgroup of G which is locally nilpotent and if the Sytow p-subgroup of M is finite or regular for p = char K, p 4= 2, while the Sylow 2-subgroup is nilpotent Of class at most 2, then G is solvable ([I0], 12,8). Wehrfritz says that the hypothesis on the Sylow p-subgroup of M, for p char K, is probably redundant: we prove this in Section 3, Some of the above results were extended to locally finite groups by Bruno and Schuur (see [1]),

Influence of normality conditions on almost minimal subgroups of a finite group

Rational rigidity and the sporadic groups

Maximal subgroups in composite finite groups

Let [Formula: see text] be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if [Formula: see text] is a normal subgroup of a finite group [Formula: see text] then the image of an [Formula: see text]-maximal subgroup [Formula: see text] of [Formula: see text] in [Formula: see text] is not, in general, [Formula: see text]-maximal in [Formula: see text]. We say that the reduction [Formula: see text]-theorem holds for a finite group [Formula: see text] if, for every finite group [Formula: see text] that is an extension of [Formula: see text] (i.e. contains [Formula: see text] as a normal subgroup), the number of conjugacy classes of [Formula: see text]-maximal subgroups in [Formula: see text] and [Formula: see text] is the same. The reduction [Formula: see text]-theorem for [Formula: see text] implies that [Formula: see text] is [Formula: see text]-maximal in [Formula: see text] for every extension [Formula: see text] of [Formula: see text] and every [Formula: see text]-maximal subgroup [Formula: see text] of [Formula: see text]. In this paper, we prove that the reduction [Formula: see text]-theorem holds for [Formula: see text] if and only if all [Formula: see text]-maximal subgroups of [Formula: see text] are conjugate in [Formula: see text] and classify the finite groups with this property in terms of composition factors.