Characterization of projective special linear groups in dimension three by their orders and degree patterns

Document Type: Research Paper

Authors

Shahrekord University

Abstract

The prime graph $\Gamma(G)$ of a group $G$ is
a graph with vertex set $\pi(G)$, the set of primes dividing the
order of $G$, and two distinct vertices $p$ and $q$ are adjacent
by an edge written $p\sim q$ if there is an element in $G$ of
order $pq$. Let $\pi(G)=\{p_{1},p_{2},...,p_{k}\}$. For
$p\in\pi(G)$, set $deg(p):=|\{q \in\pi(G)| p\sim q\}|$, which is
called the degree of $p$. We also set
$D(G):=(deg(p_{1}),deg(p_{2}),...,deg(p_{k}))$, where
$p_{1}<p_{2}<...<p_{k}$, which is called degree pattern of $G$.
The group $G$ is called $k$-fold OD-characterizable if there exists
exactly $k$ non-isomorphic groups $M$ satisfying conditions
$|G|=|M|$ and $D(G)=D(M)$. In particular, a $1$-fold
OD-characterizable group is simply called OD-characterizable. In
this paper, as the main result, we prove that projective special
linear group $L_{3}(2^{n})$ where $n\in\{4,5,6,7,8,10,12\}$ is
OD-characterizable. 

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