I'm taking an introductory group theory class, and one of the problems I was assigned is to show that if an element $g \in G$ has order $2$, then $g \in Z(G)$, where $Z(G)$ is the center of G. Can anyone help me out on this one? I'm a little stuck.
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1This isn't true, can you give more context? – Matthew Towers Mar 20 '20 at 21:55
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1See: prove that if a group contains exactly one element of order 2, then that element is in the center. – amWhy Mar 20 '20 at 21:55
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This is only true in a group with one and only one element of order 2. See the link above. – amWhy Mar 20 '20 at 21:56
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1This is clearly false, take $G=S_3$, then elements is order 2 are the transpositions, while $Z(G)$ is trivial. Didn't you miss some important assumptions on $G$? – lisyarus Mar 20 '20 at 21:57
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"In every group G in which one and only one element $x\in G$ has order 2, $x\in Z(G)$ – amWhy Mar 20 '20 at 22:00