# Group acts on left coset space of subgroup by left multiplication

From Groupprops

This article describes a pivotal group action: a group action on a set closely associated with the group. This action is important to understand and remember.

View other pivotal group actions

## Statement

Suppose is a group and is a subgroup of . Consider the left coset space , i.e., the collection of left cosets of in :

.

Then there exists a natural group action of on by left multiplication:

.

In other words, left multiplying any left coset by an element of the group yields a left coset, and this defines an action of the group on the left coset space by left multiplication. Further, the following facts are true about this group action:

- The stabilizer of the coset is the conjugate subgroup .
- The intersection of all stabilizers, which is also the kernel of the homomorphism associated with this action, is the intersection of all conjugate subgroups of in . This is termed the normal core of in .