Sums of prime element orders in 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...

Finite groups with $X$-quasipermutable subgroups of prime power order

Let $H$, $L$ and $X$ be subgroups of a finite group$G$. Then $H$ is said to be $X$-permutable with $L$ if for some$xin X$ we have $AL^{x}=L^{x}A$. We say that $H$ is emph{$X$-quasipermutable } (emph{$X_{S}$-quasipermutable}, respectively) in $G$ provided $G$ has a subgroup$B$ such that $G=N_{G}(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (with all Sylowsubgroups, respectively) $...

Sums of Dilates in Groups of Prime Order

We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or −1 and if A ⊂ Z/pZ is not too large (with respect to p), then |A+t·A| > (2+θt)|A|−w(t) for some constant w(t) depending only on t and for some explicit real number θt > 0 (except in the case |t| = 3). In the important case |t| = 2,...