Element orders and Sylow structure of finite groups

Element orders and Sylow structure of finite groups

Finite Groups with p-Sylow Coverings

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...

Invariant Sylow subgroups and solvability of finite groups

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3. Mathematics Subject Classification (2010). Primary 20D20; Secondary 20D45.