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Problem 102 (Subgroups Of Index 2 Are Normal)

Suppose that $H$ is a subgroup of $G$ with index $|G:H|=2$. Recall that the index of $H$ in $G$ is the number of distinct cosets of $H$ in $G$.

  1. Build a surjective homomorphism from $G$ to $\mathbb{Z}_2$.
  2. Show that $H$ is normal in $G$.

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