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:

  1. The operation "\(\circ\)" is well-defined, (i.e. it does not depend on the particular choice of representatives \(a_1,a_2\)).
  2. The operation "\(\circ\)" is associative.
  3. \((G/N,\circ )\) is not empty and contains at least the neutral element of \((G/N,\circ )\) is \(N\).
  4. The inverse element of an element \(aN\) of \((G/N,\circ )\) is \(a^{-1}N\).
  5. The inverse element of an element \(aN\) of \((G/N,\circ )\) is \(a^{-1}N\).

Proofs: 1

Proofs: 1 2
Theorems: 3

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013