Show that H is normal subgroup of G.

Question

Let H be a proper subgroup of G and a $\in$ G, a $\notin$ H. Suppose that for all b $\in$ G, either b $\in$ H, or Ha = Hb. Show that H is normal subgroup of G. How do I proceed on this?

Answer 1

The index of $H$ in $G$ is two that is it has exactly two right/left cosets, specifically $$ \{H, Ha\}$$ $$ \{H , a'H\} $$

Since the right/left cosets define equivalence classes $$ Ha = G \setminus H = a'H$$

Hence every left coset of $H$ is a right coset of $H$ in $G$ whence $H \triangleleft G $.

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