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. Congruence condition relating number of subgroups in maximal subgroups and number of subgroups in the whole group

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}, Lemma 1.4, Page 312