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Keywords:
derived subgroup; simple group
Summary:
We show that a finite nonabelian characteristically simple group $G$ satisfies $n=|\pi (G)|+2$ if and only if $G\cong A_5$, where $n$ is the number of isomorphism classes of derived subgroups of $G$ and $\pi (G)$ is the set of prime divisors of the group $G$. Also, we give a negative answer to a question raised in M. Zarrin (2014).
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