What is a quotient group

1. Jul 23, 2014

Greg Bernhardt

Definition/Summary

A quotient group or factor group is a group G/H derived from some group H and normal subgroup H.

Its elements are the cosets of H in G, and its group operation is coset multiplication.

Its order is the index of H in G, or order(G)/order(H).

Equations

Extended explanation

Coset multiplication of cosets g1H and g2H yields the coset g1g2H. Proof:

Multiply every element of the two cosets together:
$\{ g_1 h_1 g_2 h_2 : h_1 , h_2 \in H \}$

By self-conjugacy, we get
$\{ g_1 g_2 h_3 h_2 : h_3 , h_2 \in H \}$

where each h3 need not equal the h1 it was derived from. By closure of H, we get
$\{ g_1 g_2 h : h \in H \}$

or the coset g1g2H.

There are two trivial cases:
H is identity group -> G/H is isomorphic to G
H = G -> G/H is the identity group

The simplest nontrivial case is for where H has half the number of elements of G. It has one coset, G - H, which is both a left and a right coset, making H a normal subgroup for every possible H with that order. Its coset multiplication table is
H * H = H
H * (G-H) = (G-H)
(G-H) * H = (G-H)
(G-H) * (G-H) = H

This shows that G/H is Z(2), the 2-element cyclic group.

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