1.74: Normal subgroup
A subgroup H of a group G is normal in G (H \(\triangleleft\) G) if gH = Hg for any g ∈G. Equivalently, H ⊂ G is normal if and only if gHg -1 = H for any g ∈G, i.e., if and only if each conjugacy class of G is either entirely inside H or entirely outside H. This is equivalent to say that H is invariant under all inner automorphisms of G.
The property gH = Hg means that left and rights cosets of H in G coincide. From this one sees that the cosets form a group with the operation g 1 H * g 2 H = g 1 g 2 H which is called the factor group or quotient group of G by H , denoted by G/H .
In the special case that a subgroup H has only two cosets in G (namely H and gH for some g not contained in H ), the subgroup H is always normal in G .