On the -length of the product of two Shmidt groups.
On the prime graphs of the automorphism groups of sporadic simple groups
In this paper as the main result, we determine finite groups with the same prime graph as the automorphism group of a sporadic simple group, except .
On the proof of a theorem of Pálfy.
On the recognizability of some projective general linear groups by the prime graph
Let be a finite group. The prime graph of is a simple graph whose vertex set is and two distinct vertices and are joined by an edge if and only if has an element of order . A group is called -recognizable by prime graph if there exist exactly nonisomorphic groups satisfying the condition . A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that is recognizable, if is an odd prime and is odd. But for even , only the recognizability...
On the total number of principal series of a finite abelian group.
On two- and three-element subsets of groups.
On zero free sets.
On zeros of characters of finite groups
For a finite group and a non-linear irreducible complex character of write . In this paper, we study the finite non-solvable groups such that consists of at most two conjugacy classes for all but one of the non-linear irreducible characters of . In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable -groups. As a corollary, we answer Research Problem in [Y. Berkovich and L. Kazarin: Finite...
On zero-sum subsequences in finite Abelian groups.
Orders of some modular linear groups.
Perfect numbers and finite groups
Periodic factorization of a finite Abelian 2-group.
Permutability of centre-by-finite groups
Let be a group and be an integer greater than or equal to . is said to be -permutable if every product of elements can be reordered at least in one way. We prove that, if has a centre of finite index , then is -permutable. More bounds are given on the least such that is -permutable.
Polynomial and Term Functions over Groups with Coprime Chief Factors.
Polynomials over the permutation group of three elements
Prime divisors of conjugacy class lengths in finite groups
We show that in a finite group which is -nilpotent for at most one prime dividing its order, there exists an element whose conjugacy class length is divisible by more than half of the primes dividing .
Prime graph components of finite almost simple groups
Primitive characters of subgroups of M-groups.
Probability that an element of a finite group has a square root
Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 - ⌊√|G|⌋/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 - 1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G.
Product decompositions of quasirandom groups and a Jordan type theorem
We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If is the minimal degree of a representation of the finite group , then for every subset of with we have . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if , then has a proper subgroup...