Coprimeness among Element Orders of Finite Groups

Element orders and Sylow structure of finite groups

The Number of Finite Groups Whose Element Orders is Given

The spectrum ω(G) of a finite group G is the set of element orders of G. If Ω is a non-empty subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups G with ω(G) = Ω and put h(G) = h(ω(G)). We say that G is recognizable (by spectrum ω(G)) if h(G) = 1. The group G is almost recognizable (resp. nonrecognizable) if 1 < h(G) < ∞ (resp. h(G) = ∞). In t...

Group Theory: Graphs Related to Element Orders of Finite Groups

There are several graphs associated with finite groups which are currently of interest in the field. This subject of this project are graphs related to the element orders of a finite group. The best known such graph is the prime graph Γ(G) of the group G and is defined as follows. The vertices are the prime numbers dividing the order (=number of elements) of the group, and two such primes are l...

Black-box recognition of finite simple groups of Lie type by statistics of element orders

Given a black-box group G isomorphic to some finite simple group of Lie type and the characteristic of G, we compute the standard name of G by a Monte Carlo algorithm. The running time is polynomial in the input length and in the time requirement for the group operations in G. The algorithm chooses a relatively small number of (nearly) uniformly distributed random elements of G, and examines th...

2 2 Se p 20 05 The Number of Finite Groups Whose Element Orders is Given

For any group G, πe(G) denotes the set of orders of its elements. If Ω is a non-empty subset of N, h(Ω) stands for the number of isomorphism classes of finite groups G such that πe(G) = Ω. We put h(G) = h(πe(G)). In this paper we show that h(P GL(2, p n)) = 1 or ∞, where p = 2 α 3 β + 1 is a prime, α ≥ 0, β ≥ 0 and n ≥ 1. In particular, we show that h(P GL(2, 7)) = h(P GL(2, 3 2)) = ∞.