Support of good lines corollary to line lemma

History
This result appears as Corollary 1.6(iii) in a paper by Jonah and Konvisser, where they prove some congruence conditions and replacement theorems.

Statement
Suppose $$p$$ is a prime number and $$G$$ is a finite $$p$$-group, i.e., a group whose order is a power of $$p$$. Suppose $$\mathcal{S}$$ is a collection of subgroups of $$G$$.

We say that two elements $$A,B \in \mathcal{S}$$ support good lines in $$\mathcal{S}$$ if every maximal subgroup of $$G$$ containing $$A \cap B$$ also contains an element of $$\mathcal{S}$$.

Suppose the following is true: there exists a maximal subgroup $$M$$ of $$G$$ containing a collection $$A_1, A_2, \dots, A_m$$ of elements of $$\mathcal{S}$$ that are all normal subgroups of $$G$$, such that for any normal subgroup $$A$$ of $$G$$ such that $$A \in \mathcal{S}$$, there exists $$i$$ such that $$A$$ and $$A_i$$ are a pair that support good lines.

For a subgroup $$H$$ of $$G$$, denote by $$n(H)$$ the number of subgroups of $$H$$ that are in $$\mathcal{S}$$. Consider the statement:

$$\mathcal{A}(H): n(H) = 0$$ or $$n(H) \equiv 1 \pmod p$$

Then, if $$\mathcal{A}(H)$$ is true for every maximal subgroup $$H$$ of $$G$$, $$\mathcal{A}(G)$$ is also true.

Facts used

 * 1) uses::Line lemma

Journal references

 * , 1.6(iii)