Group acts on left coset space of subgroup by left multiplication

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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.
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Statement

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

G/H := \{ xH \mid x \in G \}.

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

g.(xH) = (gx)H.

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:

  1. The stabilizer of the coset xH is the conjugate subgroup xHx^{-1}.
  2. The intersection of all stabilizers, which is also the kernel of the homomorphism G \to \operatorname{Sym}(G/H) associated with this action, is the intersection of all conjugate subgroups of H in G. This is termed the normal core of H in G.