Integral group ring automorphisms without Zassenhaus factorization

Every finite abelian group is a subgroup of the additive group of a finite simple left brace

An answer to a question of J.G. Thompson on some generalized characters

Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order pnq, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.

We extend the definition of a finite -group to profinite groups and give a description of profinite -groups as a triple semidirect product of two prosoluble groups with a semisimple group, extending an old result of A. M. Broshi to the profinite case. We also prove that a profinite -group with finitely generated non-trivial Fitting subgroup is metabelian-by-(finite exponent). If, in addition, G is finitely generated then it is virtually metabelian polycyclic. Communicated by Mandi Schaeffer Fry

6. Normal subgroups with abelian sylow subgroups

Variants of some of the Brauer-Fowler theorems

This paper aims to treat a study on the order of every element in the higher even, odd and prime order of group for composition. In fact, express order of a group and order of an element of a group in real numbers. Here we discuss the higher order of groups in different types of order, which will give us practical knowledge to see the applications of the composition. In order to find out the order of an element am∈G in which an= e= identity element, then find the least common multiple (i.e.(LCM))= λ) of m and n. The least common multiple of two numbers is the "smallest non-zero common number," which is a multiple of both the numbers. So O(am )= λ/m. Also, if G is a finite group, n is a positive integer, and a∈G then the order of the products na. When G is a finite group, every element must have finite order, but the converse is false. There are infinite groups where each element has finite order. Finally, find out the order of every element of a group in different types of the higher even, odd and prime order of group for composition.

Let G be a finite group with an abelian normal subgroup N. When does N have a unique conjugacy class of complements in G? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups of G closed under conjugation whose intersection equals Φ ⁢ ( G ) {\Phi(G)} . In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when Φ ⁢ ( G ) = 1 {\Phi(G)=1} , these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

ABSTRACTLet G be a finite group and L(G) denotes the absolute center of G. An automorphism α of G is called an autocentral automorphism if x−1α(x)∈L(G) for each x∈G. An automorphism α of G is called a central automorphism if x−1α(x)∈Z(G) for each x∈G. An automorphism α of G is called an IA-automorphism if for each x∈G. In this paper, we find necessary and sufficient conditions on G such that every autocentral automorphism is inner. Also we characterize finite non-Abelian p-groups for which every central automorphism is IA-automorphism.

In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio [Formula: see text]. This extends the case [Formula: see text] studied in previous papers [R. Brandl and W. Shi, Finite groups whose element orders are consecutive integers, J. Algebra 143 (1991) 388–400; Y. Feng, Finite groups whose abelian subgroup orders are consecutive integers, J. Math. Res. Exp. 18 (1998) 503–506; W. Shi, Finite groups whose proper subgroup orders are consecutive integers, J. Math. Res. Exp. 14 (1994) 165–166].

We distinguish 2-maximal and strictly 2-maximal subgroups of a finite group and give examples of soluble and simple groups in which every 2-maximal subgroup is strictly 2-maximal. Let M be a maximal subgroup of a group G. We prove that every maximal subgroup of M is strictly 2-maximal in G if M is normal in G or if G is p-soluble and We describe the structure of a finite group in which all 2-maximal subgroups are Hall subgroups. In particular, it has a Sylow tower and all its Sylow subgroups are elementary abelian.

It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺. The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park.

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two main theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or possess an abelian subgroup of index 2. For our second theorem, we show that if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A_5 shows that this result is best-possible. Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.

Nilpotent decomposition in integral group rings

The normalizer problem of integral group rings has been studied extensively in recent years due to its connection with the longstanding isomorphism problem of integral group rings. Class-preserving Coleman automorphisms of finite groups occur naturally in the study of the normalizer problem. Let G G be a finite group with a nilpotent subgroup N N . Suppose that G / N G\hspace{-0.0em}\text{/}\hspace{-0.0em}N acts faithfully on the center of each Sylow subgroup of N N . Then it is proved that every class-preserving Coleman automorphism of G G is an inner automorphism. In addition, if G G is the product of a cyclic normal subgroup and an abelian subgroup, then it is also proved that every class-preserving Coleman automorphism of G G is an inner automorphism. Other similar results are also obtained in this article. As direct consequence, the normalizer problem has a positive answer for such groups.