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16.106 (2006)

Solved

Let $\pi_e(G)$ denote the set of orders of elements of a group $G$, and $h(\Gamma)$ the number of non-isomorphic finite groups $G$ with $\pi_e(G) = \Gamma$. Do there exist two finite groups $G_1, G_2$ such that $\pi_e(G_1) = \pi_e(G_2)$, $h(\pi_e(G_1)) < \infty$, and neither of the two groups $G_1, G_2$ is isomorphic to a subgroup or a quotient of a normal subgroup of the other?

Contributor: W. J. Shi

Progress

Yes, there do: for example, $G_1 = L_{15}(2^{60}).3$ and $G_2 = L_{15}(2^{60}).5$ (M. A. Grechkoseeva, Algebra and Logic, 47, no. 4 (2008), 229–241).

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