1.42: Factor group
Let N be a normal subgroup of a group G. The factor group or quotient group G/N is the set of all left cosets of N in G, i.e.:
- \(G/N = \{ aN : a \isin G \}.\)
For each aN and bN in G/N, the product of aN and bN is (aN)(bN), which is still a left coset. In fact, because N is normal:
- (aN)(bN) = a(Nb)N = a(bN)N = (ab)NN = (ab)N.
The inverse of an element aN of G/N is a -1 N.