Lemma: Factor Groups
Let \( ( G ,\ast) \) be a group and let \(N\unlhd G\) be its normal subgroup. The quotient set \(G/N:=\{aN: a\in G\}\) of all (left) cosets, together with the binary operation "\(\circ\)":
\[(a_1 N)\circ(a_2N):=(a_1\ast a_2)N\quad\quad(a_1,a_2\in G)\]
forms a group \((G/N,\circ )\), called the factor group of \(G\) with respect to the normal subgroup \(N\), (also pronounced as "G modulo N"). In particular:
 The operation "\(\circ\)" is welldefined, (i.e. it does not depend on the particular choice of representatives \(a_1,a_2\)).
 The operation "\(\circ\)" is associative.
 \((G/N,\circ )\) is not empty and contains at least the neutral element of \((G/N,\circ )\) is \(N\).
 The inverse element of an element \(aN\) of \((G/N,\circ )\) is \(a^{1}N\).
 The inverse element of an element \(aN\) of \((G/N,\circ )\) is \(a^{1}N\).
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2
Theorems: 3
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References
Bibliography
 Kramer Jürg, von Pippich, AnnaMaria: "Von den natürlichen Zahlen zu den Quaternionen", SpringerSpektrum, 2013