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Normal subgroups

Let X be a nonempty subset of a group G.

If G = <X> and H is a subgroup of G, show that H is the normal subgroup of G if and only if x^-1Hx contained in H for all x belonging to X.

ALSO show that <X> is normal in G if and only if gXg^-1 contained in <X> for all g belonging to G.

Solution Summary

This is a proof regarding normal subgroups.