Subnormalizing group intersection problem

Description

Given data

Our universe is some group $U$ (such as a linear group or a permutation group) in which products and inverses can be readily computed.

Two groups $G_{1}$ and $G_{2}$ in $U$ are specified by means of their generating sets $A_{1}$ and $A_{2}$ . We are also given that $G_{1}$ is a subnormalizing subgroup for $G_{2}$ , or in other words, $G_{1}$ is a subnormal subgroup in the subgroup generated by $G_{1}$ and $G_{2}$ .

Goal

We are required to determine a generating set for $G_{1} \cap G_{2}$ .

Solution

Idea

The idea is to use the solution approach that we already saw for the normalizing group intersection problem, by the fact that every subnormal subgroup has a subnormal series. There are thus two steps:

Write a subnormal series for $G_{1}$ in the subgroup $< G_{1}, G_{2} >$ , say something of the form:

$G_{1} = K_{r} ◃ K_{r - 1} ◃ K_{r - 2} ◃ \dots ◃ K_{0} = < G_{1}, G_{2} >$

Suppose inductively that we have computed $M_{i} = G_{2} \cap K_{i}$ . Then we can try computing $G_{2} \cap K_{i + 1}$ by computing $M_{i} \cap K_{i + 1}$ . To do this, appeal to the normalizing group intersection problem.