# Collection of groups satisfying a universal non-divisibility condition

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## Definition

Suppose is a collection of finite -groups, i.e., groups of prime power order for a prime number . We say that satisfies a universal non-divisibility condition if the following equivalent conditions hold:

- For any finite -group containing a subgroup isomorphic to an element of , the number of subgroups of isomorphic to elements of is
*not*divisible by . - For any finite -group containing a subgroup isomorphic to an element of , the number of subgroups of isomorphic to elements of is
*not*divisible by . - For any finite -group and any normal subgroup of containing a subgroup isomorphic to an element of , the number of normal subgroups of isomorphic to elements of and contained in is
*not*divisible by . - For any finite -group that contains a subgroup isomorphic to an element of , the number of p-core-automorphism-invariant subgroups of is
*not*divisible by . - For any finite group containing a subgroup isomorphic to an element of , the number of subgroups of isomorphic to an element of is
*not*divisible by .