Proving congruence conditions on number of subgroups

This article discusses techniques to prove that the number of subgroups of a given order, or given isomorphism type, or satisfying certain conditions, satisfies a congruence condition: typically a condition of the form that it is $$1$$ mod $$p$$, where the subgroups under consideration are $$p$$-groups.

There is a huge array of techniques for proving such results. We first discuss how, for $$p$$-groups, we can typically reduce the problem to the case where the whole group is itself a $$p$$-group, and further, that the congruence condition on the number of subgroups is equivalent to the same congruence condition on the number of normal subgroups. Next, we show that such problems typically break up into two parts: showing existence (i.e., showing that there exists one subgroup) and then proving the congruence condition modulo existence. Then, we discuss the common tools used: group actions, the use of projective spaces, and the use of inductive arguments based on maximal subgroups.