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- W3160848507 abstract "Let $G$ be a group, define an equivalence relation $sim$ as below: $$forall g, h in G, g sim h Longleftrightarrow |g| = |h|$$ the set of sizes of equivalence classes with respect to this relation is called the same-order type of $G$ and denoted by $alpha(G)$. And $G$ is said a $alpha_n$-group if $|alpha(G)|=n$. Let $pi(G)$ be the set of prime divisors of the order of $G$. A simple group of $G$ is called a simple $K_n$-group if $|pi(G)|=n$. We give a new characterization of simple $K_3$-groups using same-order type. Indeed we prove that a nonabelian simple group $G$ has same-order type {r, m, n, k, l} if and only if $G cong PSL(2,q)$, with $q=7, 8$ or $9$. This result generalizes the main results in cite{KKA}, cite{Sh} and cite{TZ1}. Motived by the main result in cite{TZ1} L. J. Taghvasani and M. Zarrin put the following Conjecture 2.10: textit{Let $S$ be a nonabelian simple $alpha_n$-group and $G$ a $alpha_n$-group such that $|S|=|G|$. Then $S cong G$}. In this paper with a counterexample we give a negative answer to this question." @default.
- W3160848507 created "2021-05-24" @default.
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- W3160848507 date "2019-11-30" @default.
- W3160848507 modified "2023-09-22" @default.
- W3160848507 title "A new characterization of simple $K_3$-groups using same-order type" @default.
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