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Suppose p is a prime and T is a non-identity finite p -group. We show that if p is odd, then there exist two non-identity characteristic subgroups K 1 (T) and K 2 (T) of T such that K 1 and K 2 jointly control transfer in every finite group G containing T as a Sylow p -subgroup, in the sense that the normalizers of K 1 (T) and K 2 (T) in G determine the largest abelian factor group of G that is a p -group. We also give a new example of a characteristic subgroup K (T) of T such that K controls transfer in G by itself if p ⩾ 5. [ABSTRACT FROM AUTHOR]

Suppose p is a prime and T is a non-identity Sylow p -subgroup of a finite group G for which C G (O p (G)) ⩽ O p (G). D. Bundy, N. Hebbinghaus, and B. Stellmacher determined the structure of G in an important case in which C G (Ω 1 (Z (T))) and N G (B e (T)) generate a proper subgroup of G , where B e (T) is the Baumann subgroup of T according to a particular definition. We apply their theorem to prove an analogous theorem that uses a different definition of the Baumann subgroup. This result is applied in a related article to prove a transfer theorem for odd primes. [ABSTRACT FROM AUTHOR]

We consider two variants of those Abelian groups with all proper strongly invariant subgroups isomorphic and give an in-depth study of their basic and specific properties in either parallel or contrast to the Abelian groups with all proper fully invariant (respectively, characteristic) subgroups isomorphic, which are studied in details by the current authors in Commun. Algebra (2015) and in J. Commut. Algebra (2023). In addition, we also explore those Abelian groups having at least one proper strongly invariant subgroup isomorphic to the whole group. [ABSTRACT FROM AUTHOR]

The automorphism group of a Dihedral group of order 2n is isomorphic to the holomorph of a cyclic group of order n. The holomorph of a cyclic group of order n is a complete group when n is odd. Hence automorphism groups of Dihedral groups of order 2n are its own automorphism groups whenever n is odd. In this paper, we prove that the result is also true for those Dihedral groups of order 2n where n is twice a prime number. [ABSTRACT FROM AUTHOR]

In this paper, first we give a necessary and sufficient condition on a non-abelian p-group G of exponent p 2 (p > 2) possessing a unique proper non-trivial characteristic subgroup. Then we study some families of finite groups with a unique proper non-trivial characteristic subgroup. In particular, we study metacyclic groups, 2-generated p-groups, minimal non-abelian groups, A 2 -group and also p-groups of order less than p 7 . [ABSTRACT FROM AUTHOR]

In this article, we classify some finite p-groups whose normal subgroups are characteristic. Particularly we restrict our attention to all finite p-groups of maximal class with an abelian maximal subgroup, finite 3-groups of maximal class, finite split metacyclic p-groups, minimal non-abelian p-groups and all p-groups of maximal class of order less than p6. [ABSTRACT FROM AUTHOR]

It is well-known that every virtually abelian group contains an abelian characteristic subgroup of finite index. We shall say that a group class is F-characteristic if any group containing an -subgroup of finite index has also a characteristic subgroup of finite index that belongs to. Thus the class of abelian groups is F -characteristic. Many other relevant group classes have recently been proved to be F-characteristic, and the aim of this paper is to give a further contribution to this topic, in particular in the context of classes of generalized nilpotent groups. [ABSTRACT FROM AUTHOR]

It is known that if a group G has an abelian subgroup of finite index n, then it contains an abelian characteristic subgroup A of index at most nn. The aim of this paper is to improve this bound by showing that the characteristic subgroup A can be chosen of index at most n2. Examples prove that this bound is the best possible. Our main result is obtained as an application of a general method for the construction of large characteristic subgroups. [ABSTRACT FROM AUTHOR]

In this paper, we define the concept of integral for an arbitrary group, and we try to answer this question: Which groups can be considered as a commutator subgroup? For instance, we will show that symmetric groupSnforn≥ 3, generalized quaternion groupforn≥ 4 and Dihedral groupDnforn≥ 3 cannot occur as commutator subgroup of any group. Moreover, we characterize all finite groups with the cyclic commutator subgroup isomorphic toor. [ABSTRACT FROM PUBLISHER]

The concept of autopermutable subgroups was introduced by Housieni and Moghaddam in 2014. In the present article, we study the properties of auto- permutable subgroups of a given group and among other results, it is shown that how auto-permutability induces characteristic properties. Also we show that all subgroups of a Hamiltonian nilpotent group always enjoy this property. [ABSTRACT FROM AUTHOR]

In this paper, finite groups having exactly two conjugate classes of non-subnormal subgroups ofGare completely classified. [ABSTRACT FROM AUTHOR]

Fusion class of a group consists of all the elements which are related by the fusion relation, that is; the elements which are conjugate in their automorphism group. In this paper we study the structure of groups in which union of fusion classes with identity element 1 becomes a subgroup. [ABSTRACT FROM AUTHOR]

Abstract: We define a new non-identity characteristic subgroup, , of a finite p-group S, and prove that is characteristic in a significant family of p-stable finite groups G having S as a Sylow p-subgroup. This result is analogous to an earlier result about by the first author and has a very short proof. [Copyright &y& Elsevier]

Abstract: Let G be a locally finite group having a subgroup N of finite index n that possesses a normal series each of whose quotients is either locally nilpotent or satisfies an outer commutator law . We show that G contains a characteristic subgroup H of finite index that has a characteristic series with the same properties. Moreover, the index of H in G is bounded by a function depending only on n, k and the weight of the . [Copyright &y& Elsevier]

Abstract: As a counterpart for the prime 2 to Glauberman''s ZJ-theorem, Stellmacher proves that any nontrivial 2-group S has a nontrivial characteristic subgroup with the following property. For any finite -free group G, with S a Sylow 2-subgroup of G and with self-centralizing, the subgroup is normal in G. We generalize Stellmacher''s result to fusion systems. A similar construction of can be done for odd primes and gives rise to a Glauberman functor. [Copyright &y& Elsevier]

Abstract: We consider structures consisting of an abelian group with a subgroup distinguished by a 1-ary relation symbol , and complete theories of such structures. Such a theory is -categorical if has models of cardinality with , and given any two such models with , there is an isomorphism from to which is the identity on . We classify all complete theories of such structures in terms of the cardinal pairs in which they are categorical. We classify algebraically the of finite order with of order which are ()-categorical. [Copyright &y& Elsevier]

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. [ABSTRACT FROM AUTHOR]