Jonah-Konvisser line lemma

History
This lemma was proved by Jonah and Konvisser in their 1975 paper. It was a basic lemma to their proofs of the Jonah-Konvisser abelian-to-normal replacement theorem and Jonah-Konvisser elementary abelian-to-normal replacement theorem.

Statement
Suppose $$p$$ is a prime number. Suppose $$G$$ is a finite $$p$$-group, and $$\mathcal{S}$$ is a collection of proper subgroups of $$G$$. For any subgroup $$H$$ of $$G$$, let $$n(H)$$ denote the number of elements of $$\mathcal{S}$$ inside $$H$$.

For a subgroup $$H$$ of $$G$$, consider the statement $$A(H)$$:

$$A(H)$$: Either $$n(H) = 0$$ or $$n(H) \equiv 1 \pmod p$$.

Suppose there is an origin for $$\mathcal{S}$$ in $$G$$: there is a maximal subgroup $$M$$ of $$G$$ such that whenever $$N$$ is a maximal subgroup of $$G$$ containing an element of $$\mathcal{S}$$, every maximal subgroup of $$G$$ containing $$M \cap N$$ contains an element of $$\mathcal{S}$$.

Then, if $$A(M)$$ holds for every maximal subgroup $$M$$ of $$G$$, $$A(G)$$ holds.

Facts used

 * 1) uses::Congruence condition relating number of subgroups in maximal subgroups and number of subgroups in the whole group