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- W1904652034 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 $psim q$. Let $|G= p_1^{alpha_1}cdot p_2^{alpha_2}cdots p_k^{alpha_k}$, where $p_1<p_2 <dots < p_k$ are primes. For $pin pi(G)$, let $deg(p) = |{qin pi(G)|psim q}|$ be the degree of $p$ in the graph $Gamma(G)$, we define $D(G)=(deg(p_1),deg(p_2),dots,deg(p_k))$ 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 $Aut(L)equiv Z_4$ 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." @default.
- W1904652034 created "2016-06-24" @default.
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- W1904652034 date "2014-12-01" @default.
- W1904652034 modified "2023-09-24" @default.
- W1904652034 title "OD-characterization of $S_4(4)$ and its group of automorphisms" @default.
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