Abstract: Let $p$ be an odd prime, and let $S$ be a $p$-group with a unique elementary abelian subgroup $A$ of index $p$. We classify the simple fusion systems over all such groups $S$ in which $A$ is essential. The resulting list, which depends on the classification of finite simple groups, includes a large variety of new, exotic simple fusion systems.

Abstract: We study finite groups in which each primary subgroup is self-normalizing or \(\mathfrak{U}\)-subnormal in the class U of all supersoluble groups. In particular, these groups have a Sylow tower.

Abstract: We prove that every non-abelian finite simple group is generated by an involution and an element of prime order.

Abstract: We describe structure of quasihomomorphisms from arbitrary groups to discrete groups. We show that all quasihomomorphisms are “constructible”, i.e., are obtained via certain natural operations from homomorphisms to some groups and quasihomomorphisms to abelian groups. We illustrate this theorem by describing quasihomomorphisms to certain classes of groups. For instance, every unbounded quasihomomorphism to a torsion-free hyperbolic group H is either a homomorphism to a subgroup of H or is a quasihomomorphism to an infinite cyclic subgroup of H.

Abstract: The investigation of finite groups classifying as “extreme” according to certain quantitative conditions has a long history. The aim of this paper is to classify the pairs (G, α) where G is a finite group and α is an automorphism of G with a cycle of length greater than . In particular, it is shown that every finite group G having such an automorphism is abelian.

Abstract: In this paper, we prove that a finite group with a splitting automorphism of odd order is solvable. By using this result, we prove that a locally finite group with a splitting automorphism of odd order is locally solvable.

Abstract: For a group G, we say that x, y ∈ G are in the same z-class if their centralizers in G are conjugate. The notion of z-class has origin in a connection between geometry and groups. However, as the notion is purely group theoretic, in this paper, we focus our attention on the influence of the z-classes on the structure of the group. The number of z-classes is invariant for a family of isoclinic groups. We obtain bounds for the number of z-classes in certain families of groups. A non-abelian finite p-group contains at least p + 2z-classes. Moreover, we characterize the non-abelian p-groups with p + 2z-classes; these are precisely, up to isoclinism, the p-groups of maximal class with an abelian subgroup of index p.

Abstract: Let G be a non-abelian group of order 2n which has a cyclic subgroup of index 2 and let U ( [G]) be the unit group of the group algebra [G] of G over the field containing q = pkelements. In this paper, recurrence relations are established to determine the structure of the unit group of [G] when p > 2.

Abstract: In this article, we consider a type of generalized group identity and extend some earlier results. For example, we show that, if D is a division ring with infinite center, then every subnormal subgroup of GLn(D) satisfying a generalized group identity over GLn(D) is central.

Abstract: In this paper, (1) we deduce a formula for the number of automorphism of finite Abelian p-group of rank two and (2) a formula for the number of auto-morphism of finite Abelian group of rank two Zm×Zn, by using partition of number of cyclic subgroups of group .

Abstract: Let $p>3$ be a prime. For each maximal subgroup $H\leqslant\mathrm{GL}(d,p)$ with $|H| \geqslant p^{3d+1}$, we construct a $d$-generator finite $p$-group $G$ with the property that $\mathrm{Aut}(G)$ induces $H$ on the Frattini quotient $G/\Phi(G)$ and $|G| \leqslant p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovacs. The groups $G$ that we exhibit have exponent $p$, and of all such groups $G$ with the desired action of $H$ on $G/\Phi(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.

Abstract: We prove that every 2-subgroup of a periodic group saturated with groups of Lie type over fields of odd characteristics whose Lie ranks are bounded as a whole is Chernikov. In particular, every such group is locally finite.

Abstract: Let G be a finite group. We say that a subgroup H of G is weakly SΦ-supplemented in G if G has a subgroup T such that G = HT and H∩T ≤ Φ(H)HsG, where HsG is the subgroup of H generated by all those subgroups of H that are s-permutable in G. In this paper, we investigate the influence of weakly SΦ-supplemented subgroups on the structure of finite groups. Some new characterizations of p-nilpotency and supersolubility of finite groups are obtained.

Abstract: The main aim of this paper is to investigate two characteristic subgroups ω𝒜(G) and θ𝒜(G) of a finite group G, which are defined as the intersections of the normalizers of derived subgroups of subnormal and non-subnormal subgroups of G respectively. Our main theory improve and extend some earlier results.

Abstract: We find finite almost simple groups with prime graphs all of whose connected components are cliques, i.e., complete graphs. The proof is based on the following fact, which was obtained by the authors and is of independent interest: the prime graph of a finite simple nonabelian group contains two nonadjacent odd vertices that do not divide the order of the outer automorphism group of this group.

Abstract: Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free subgroup of the mapping class groups.

Abstract: Two basic results on S-rings over an Abelian group are the Schur theorem on multipliers and the Wielandt theorem on primitive S-rings over groups with a cyclic Sylow subgroup. Neither of these is directly generalized to the non-Abelian case. Nevertheless, we prove that the two theorems are true for central S-rings over any group, i.e., for S-rings that are contained in the center of the group ring of that group (such S-rings arise naturally in the supercharacter theory). Extending the concept of a B-group introduced by Wielandt, we show that every Camina group is a generalized B-group, whereas simple groups, with few exceptions, cannot be of this type.

Abstract: In this paper we define Autocamina Group. In Camina group, conjugacy class of an element outside the commutator subgroup coincides with the coset of the commutator subgroup. Similarly we call a group an Autocamina group, if fusion class is the coset of autocommutator subgroup. We study the structure of Autocamina groups in this paper.

Abstract: In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group $G$. This quantity measures the probability of two random cyclic subgroups of $G$ commuting. Explicit formulas are obtained for some particular classes of groups. A criterion for a finite group to be an Iwasawa group is also presented.

Abstract: Finite groups whose prime graphs do not contain triangles are investigated. In the present part of the study, the isomorphic types of prime graphs and estimates of the Fitting length of solvable groups are found and almost simple groups are determined.

Abstract: We study finite groups with given indices of 2-maximal subgroups. Suppose that every 2-maximal subgroup H of a group G has the property: if H ≤ U ≤ V ≤ G, then |V :U| divides the square of a prime. Then G is solvable.

Abstract: A subgroup H of a finite group G is called c*-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is S-quasinormally embedded in G. In this paper, we investigate the local c*-supplementation of maximal subgroups of some Sylow p-subgroup and present some sufficient and necessary conditions for a finite group to be p-nilpotent. As applications, we give some sufficient conditions for a finite group to be in a saturated formation.

Abstract: man) ABSTRACT. A formula, analogous to the classical Burnside lemma, is developed which counts orbit representatives from a set under a group ac- tion with a given stabilizer subgroup conjugate class. This formula is applied in a manner analogous to a proof of Polya's theorem to obtain an enumeration of patterns with a given automorphism group.