What is the Definition and Explanation of a Quotient Group?

[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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[fresh_42]

A good exercise which provides insights is the following:
Prove that a subgroup is normal if and only if its quotient has a group structure.

Given a group $G$ and a subgroup $U<G$. Then we can always consider the set $G/U=\{\,gU\,|\,g\in G\,\}$ of equivalence classes with respect to $U$. But $G/U$ is only a group itself, if $U \triangleleft G$ is a normal subgroup.