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Some new compatible groups
Zhaochen Ding, Gabriel Verret
https://arxiv.org/abs/2509.17330 https://arxiv.org/pdf/2509.17330…

Some new compatible groups
Two finite groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. We prove a new sufficient condition for two groups to be compatible. As a corollary, we obtain that nilpotent groups of the same order are compatible, and so are groups of the same square-free order.