= Coset action
{title2=$g\cdot(aH)=(ga)H$}
{wiki}
A group acts on the left cosets of a subgroup by left multiplication; its kernel is the core of that subgroup.
Let $H\leq G$ be a <subgroup>. The notation $G/H$ here means the <set> of left <cosets> $\{aH:a\in G\}$, even when $H$ is not a <normal subgroup>. Define
$$
g\cdot(aH)=(ga)H,\qquad
\rho_H:G\longrightarrow\operatorname{Sym}(G/H).
$$
This gives a <group action> without needing a <quotient group> structure on $G/H$.<Group Actions by Keith Conrad>[Example 2.12, printed pages 7–8: left multiplication on left cosets, without normality]{ref}
The representative check is essential: if $aH=bH$, then $a=bh$ for some $h\in H$, and $(ga)H=(gbh)H=(gb)H$. Thus equal input <cosets> give equal output <cosets>. The <identity element> fixes every <coset>, and <associativity> gives $g_1\cdot(g_2\cdot aH)=(g_1g_2)\cdot aH$. Each induced map is a <permutation>, with inverse induced by $g^{-1}$. This is a <transitive group action> because $ba^{-1}$ sends $aH$ to $bH$.
The <stabilizer subgroup> of $H$ is $H$ itself. At a general <coset>, the <stabilizer subgroup> and the <group kernel> are
$$
G_{aH}=aHa^{-1},\qquad
\ker\rho_H=\bigcap_{a\in G}aHa^{-1}=\operatorname{Core}_G(H).
$$
Consequently the <coset action> is a <faithful group action> exactly when the <subgroup core> is trivial.<Group Theory (v4.01) by James S. Milne>[Chapter 4, Action on the left cosets, printed page 64, formulas preceding Remark 4.22 and Remark 4.22(a)]{ref}
Indeed, $g\cdot(aH)=aH$ exactly when $a^{-1}ga\in H$, or $g\in aHa^{-1}$. Intersecting these membership conditions over all <cosets> gives the <group kernel>. <Conjugation> permutes the <subgroups> in this intersection, making it a <normal subgroup> of $G$ contained in $H$. Conversely, any <normal subgroup> of $G$ contained in $H$ lies in every $aHa^{-1}$. This explains why the <subgroup core> is the largest such <normal subgroup>: it is exactly the portion of $H$ invisible at every label, rather than merely at the label $H$.
For a <normal subgroup> $H$, every $aHa^{-1}$ equals $H$, so $\ker\rho_H=H$. For $H=\{e\}$, the construction is the <regular group action> behind <Cayley's theorem>. Between these extremes, a nontrivial <subgroup> with trivial <subgroup core> can give a <faithful group action> on fewer labels. If the <index of a subgroup> is finite, $[G:H]=n$, the <first isomorphism theorem> gives
$$
G/\operatorname{Core}_G(H)\cong\operatorname{im}\rho_H\leq S_n.
$$
Thus a <subgroup> of small <index of a subgroup> exposes a small <permutation group> associated with $G$. The two examples below and in <nonfaithful coset action of the symmetric group on three letters> show exactly which information can survive.
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