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Let’s define $\sigma(G)$ as the sum of orders of all normal subgroups of a finite group $G$. Do there exist two finite groups $H$ and $K$ such, that $\sigma(H) = |H| + |K| = \sigma(K)$, $|H|$ is even and $|K|$ is odd?

It is quite obvious, that pair of cyclic groups $H$ and $K$ satisfies that condition iff $|H|$ and $|K|$ form an amicable pair. Thus, an even-odd amicable pair would have solved the question. But whether they do exist is an open problem!

Still, the pairs of non-cyclic non-isomorphic finite groups $H$ and $K$, that satisfy the condition $\sigma(H) = |H| + |K| = \sigma(K)$ actually do exist. For example, $D_{10}$ and $D_{19}$. However, the orders of both groups in the example are even, so it does not help us much now. And it would be interesting to know, if there is such an "even-odd pair"...

This question was inspired by: Is there an odd-order group whose order is the sum of the orders of the proper normal subgroups?

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