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A finite group $G$ is called EPO-group if every non-identity element of $G$ has prime order.

For a given finite group $G$, let $\pi (G)$ denote the set of all prime divisors of $|G|$.

Does there exist non-isomorphic EPO-groups $G_1$, $G_2$ of same order with $|\pi(G_i)|\geq 2$ , $i=1,2$ ?

2 Answers 2

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The smallest examples are the groups denoted (147,4) and (147,5) in the SmallGroups library. They are both semidirect products $Z_7^2:Z_3$.

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Yes, finite, non-isomorphic EPO-groups of the same order are possible. See DonAntonio's answer to a question in the recent thread

When will two groups be isomorphic???

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    Thank you quasi for pointing out this post, but actually I want to know some thing more, which I have now explained in this edited post. – 2017-01-17