For distinct primes p and q, the alternating group A_n, for every sufficiently large n, does not embed into any finite group invariably generated by an order-p and an order-q element.
On the diameter of intersection graphs of finite groups
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abstract
The intersection graph $\Delta_G$ of a finite group $G$ is a simple graph with vertices the non-trivial proper subgroups of $G$, and an edge between two vertices if their corresponding subgroups intersect non-trivially. These graphs were introduced by Cs\'ak\'any and Poll\'ak in 1969. In this paper we answer two long-standing open questions posed by Cs\'ak\'any and Poll\'ak concerning the diameter of intersection graphs. We prove some necessary conditions for a non-simple group to have an intersection graph of diameter 4. We also construct the first examples of non-simple groups and alternating groups whose intersection graphs have diameter 4.
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Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements
For distinct primes p and q, the alternating group A_n, for every sufficiently large n, does not embed into any finite group invariably generated by an order-p and an order-q element.