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I. Let $G$ be a group in which, for some integer $n\gt 1$, $(ab)^n=a^n b^n$ for all $a,b\in G$. Show that

  1. $G^{(n)}=\{x^{n}|x\in G\}$ is a normal subgroup of $G$.

  2. $G^{(n−1)}=\{x^{n−1}|x\in G\}$ is a normal subgroup of $G$.

II. Let $G$ be as in the problem above. Show that

  1. $a^{n−1}b^n=b^n a^{n−1}$ for all $a,b\in G$.

  2. $(aba^{−1}b^{−1})^{n(n−1)}=e$ for all $a,b\in G$.

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    These two exercises are from I.N. Herstein's Topics in Algebra (2.ed, page 54, ex 18 and ex 19).2012-02-08
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    yes the problem is from i.n.herstein ch.2 section on normal subgroups. i copied it as it is out of laziness.2012-02-09

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