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Let $(G,*)$ be an abelian group with the identity $e$. An element $a\in G$ is called an idempotent if $\,a^2 = e\,$ (where $\,a^2 = a*a).\,$ Let $S = \{a \in G\mid a^2 = e\}.$

How do I prove $S$ is a subgroup of $G$?

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    It sounds like you mean for $S$ to be a *subset*. Also, elements such that $a^2=a$ are called idempotent, but elements such that $a^2=e$ are not. If you don't really need to define that (which it looks like you don't) you should probably leave it out.2012-12-05
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    Do you want $S$ to be the subset of $G$ that consists of the idempotent elements of $G$?2012-12-05
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    @rschwieb: Wish you edit the title as well. :)2012-12-05
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    @BabakSorouh I would spend the time but I think this is probably a duplicate anyhow...2012-12-05
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    @rschwieb: Have a look at this link. http://math.stackexchange.com/q/56763/8581 when $H=\{e\}$.2012-12-05
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    [Prove that if $G$ is abelian, then $H = \{a \in G \mid a^2 = e\}$ is subgroup of $G$](http://math.stackexchange.com/q/395747), [If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$](http://math.stackexchange.com/q/146871), [Prove that $H = \{x \in G \mid x=x^{-1}\}$ is a subgroup.](http://math.stackexchange.com/q/203455)2016-10-13

4 Answers 4