m. r. darafsheh

The representation theory of finite groups has important applications in investigatingthe structure of abstract finite groups.This theory was developed by Frobenusmore than 100 years ago and later studied by Burnside, Schur, and Brauer. Inthis article we try to explain fundamental theorems of this theory and mention a fewapplications.

The Gauss quadratic reciprocity law is one of the most important theorems in number theory which Gauss proved it in nineteen years old. In this paper, we first prove some facts about the character of finite abelian groups. Then, using them, we provide a proof for the Gauss quadratic reciprocity law.

Let $G$ be a finite group. The prime graph of $G$ is a graph $Gamma(G)$ with vertex set $pi(G)$, the set of all prime divisors of $|G|$, and two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. In this paper we prove that if $Gamma(G)=Gamma(G_2(5))$, then $G$ has a normal subgroup $N$ such that $pi(N)subseteq{2,3,5}$ and $G/Nequiv G_2(5)$.

Let $L := U_3(11)$. In this article, we classify groups with the same order and degree pattern as an almost simple group related to $L$. In fact, we prove that $L$, $L:2$ and $L:3$ are OD-characterizable, and $L:S_3$ is $5$-fold OD-characterizable.