Section 8.5
Quotient groups

Here we introduce computations modulo a normal subgroup and the corresponding construction of the quotient group. The procedure is similar to the construction of a residue class ring.

Let G be a group and let N be a normal subgroup of the group G. The notions of left coset (a set of the form gN) and right coset (a set of the form Ng) of N in G coincide since normal subgroups satisfy gN = Ng for all g G. Thus, we can just speak of cosets. Normal subgroups play the same role for groups as ideals do for rings.


If X, Y are subsets of the group G and a, b G, then we write aX = {ax | x X},   XaY = {xay | x X, y Y}, aXbY = {axby | x X, y Y}, etc.


Proposition

Suppose that N is a normal subgroup of G. Then, for all a, b G,

  1. (ab)N = a(bN);
  2. aNN = aN;
  3. aNbN = abN;
  4. if a bN, then aN = bN.


Due to these properties, the set G/N of cosets admits the following group structure.

Definition

We call the group G/N with

the quotient group of G with respect to N.

The quotient group G/N is sometimes also called G mod N.