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Normal Subgroups A subgroup H of a group G is normal in G if gH = Hg for all g E G. That is, a normal subgroup of a group G is one in which the right and left cosets are precisely the same. Example 10.1. Let G be an abelian group. Every subgroup H of G is a normal subgroup. Since gh = hg for all ge G and h E H, it will always be the case that gH = Hg. Example 10.2. Let H be the subgroup of S3 consisting of elements (1) and (12). Since (123) H = {(123), (13)} and H(123) = {(123), (23) }, H cannot be a normal subgroup of $3. However, the subgroup NV, consisting of the permutations (1), (123), and (132), is normal since the cosets of NV are N = {(1), (123), (132)} (12)N = N(12) = {(12), (13), (23) }