Jonah-Konvisser origin for a collection of proper subgroups

Statement

Suppose $p$ is a prime number, and $G$ is a nontrivial finite $p$-group of order at least $p^{2}$. Suppose ${\mathcal {S}}$ is a collection of proper subgroups of $G$. (For our purposes, if ${\mathcal {S}}$ originally contained $G$ itself, we could throw it out).

An origin for $S$ is a maximal subgroup $M$ of $G$ such that if $N$ is another maximal subgroup of $G$ containing an element of ${\mathcal {S}}$, then every maximal subgroup of $G$ containing $M\cap N$ also contains an element of ${\mathcal {S}}$.

Related notions

• Jonah-Konvisser local origin for a collection of proper subgroups: Being a local origin is a fairly strong condition that implies that every maximal subgroup containing it is an origin.

Facts for which this is used

• Jonah-Konvisser line lemma

References

• Counting abelian subgroups of p-groups: a projective approach by Marc Konvisser and David Jonah, Journal of Algebra, ISSN 00218693, Volume 34, Page 309 - 330(Year 1975): ^{PDF (ScienceDirect)More info}, Definition 1.5(i)