OD-characterization of $S_4(4)$ and its group of automorphisms

Subject Areas : Group theory

P. Nosratpour ¹

1 - Department of mathematics, Ilam Branch, Islamic Azad university, Ilam, Iran

Received: 2013-01-01 Accepted : 2013-01-01 Published : 2013-09-01

Keywords: Finite simple group, OD-characterization, group of lie type,

Abstract :

Let $G$ be a finite group and $\pi(G)$ be the set of all prime divisors of $|G|$. The prime graph of $G$ is a simple graph $\Gamma(G)$ with vertex set $\pi(G)$ and two distinct vertices $p$and $q$ in $\pi(G)$ are adjacent by an edge if an only if $G$ has an element of order $pq$. In this case, we write $p\sim q$. Let $|G= p_1^{\alpha_1}\cdotp_2^{\alpha_2}\cdots p_k^{\alpha_k}$, where $p_1<p_2 <\dots < p_k$ are primes. For $p\in \pi(G)$, let $deg(p) = |\{q\in \pi(G)|p\sim q\}|$ be the degree of $p$ in the graph $\Gamma(G)$, we define $D(G)=(deg(p_1),deg(p_2),\dots,deg(p_k))$ and call it the degree pattern of $G$. A group $G$ is called $k$-fold OD characterizable if there exist exactly $k$ non-isomorphic groups $S$ such that $|G|=|S|$ and $D(G) = D(S)$. Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let $L = S_4(4)$ be the projective symplectic group in dimension 4 over a field with 4 elements. In this article, we classify groups with the same order and degree pattern as an almost simple group related to L. Since $Aut(L)\equivZ_4$ hence almost simple groups related to $L$ are $L$, $L : 2$ or $L : 4$. In fact, we prove that $L$, $L : 2$ and $L : 4$ are OD-characterizable.

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