Alg7.16

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Let G be a group such that (ab)^{2}=a^{2}b^{2} for all a,b\in G. Prove G is abelian.

abab=(ab)^{2}=a^{2}b^{2}\Rightarrow a^{{-1}}(abab)b^{{-1}}=a^{{-1}}(a^{2}b^{2})b^{{-1}}\Rightarrow ba=ab\,

Therefore, every a\in G commutes with every other element of the group.


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