We give a new proof of the well known Wedderburn's little theorem (1905) that a finite‎ ‎division ring is commutative‎. ‎We apply the concept of Frobenius kernel in Frobenius representation theorem in finite group‎ ‎theory to build a proof‎.

A G-Frobenius graph, as defined recently by Fang, Li, and Praeger, is a connected orbital graph of a Frobenius group G = K:H with Frobenius kernel K and Frobenius complement H. Γ is also shown to be a Cayley graph, Γ = Cay(K,S) for K and some subset S of the group G. On the other hand, a network N with a routing function R, written as (N,R), is an undirected graph N together with a routing R wh...

If Q is a quasigroup that is free in the class of all quasigroups which are isotopic to an abelian group, then its multiplication group MltQ is a Frobenius group. Conversely, if MltQ is a Frobenius group, Q a quasigroup, then Q has to be isotopic to an abelian group. If Q is, in addition, finite, then it must be a central quasigroup. A quasigroup Q = Q(·) is usually defined as a binary system i...

The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...

Let a and b be positive, relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In...