Definition – Quotient Group.

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1. Definition

Definition

Let \langle G,* \rangle be a group, and let H be a normal subgroup of G. We define the quotient group of G by H as the set

\frac{G}{H} = \{g*H~|~ g\in G \}

ie, the set of left (as in our definition) or right cosets of H in G (it does not matter which).

It can be shown (by applying the group axioms) that the quotient group of G by H is also a group (and hence a subgroup of G).

Theorems

  1. The function f:G\rightarrow\frac{G}{H} where

f(g) = g*H

Is a homomorphism between G and \frac{G}{H}. The proof is as follows. Consider f(g_1 * g_2), which is mapped to (g_1*g_2) *H = (g_1 *H)*(g_2 *H). But this says f(g_1 * g_2) = f(g_1)*f(g_2). \square