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If \(X\) is a subset of \(G\) such that \(xy = yx\) for all \(x,y\in X\), then \(<X>\) is Abelian.

I'm trying to understand the proof to the above statement in my book. This may be trivial but for some reason I am not seeing it.

It starts with saying [tex] X \subset C_{G}(X) [/tex] by the hypothesis and since [tex] C_{G}(X) [/tex] is a subgroup, we must have [tex] <X> \subset C_{G}(X) [/tex] and so [tex] X \subset C_{G}(<X>) [/tex]. Then, just as above we have [tex] <X> \subset C_{G}(<X>) [/tex] and so <X> is abelian as desired.

i didn't understand how the book went from saying [tex] <X> \subset C_{G}(X) [/tex] and concluding that [tex] X \subset C_{G}(<X>) [/tex]. it seems to me that [tex] X \subset C_{G}(<X>) [/tex] is clear from the hypothesis and i don't see why it was even needed to show that [tex] <X> \subset C_{G}(X) [/tex].

am i missing something here?

I'm trying to understand the proof to the above statement in my book. This may be trivial but for some reason I am not seeing it.

It starts with saying [tex] X \subset C_{G}(X) [/tex] by the hypothesis and since [tex] C_{G}(X) [/tex] is a subgroup, we must have [tex] <X> \subset C_{G}(X) [/tex] and so [tex] X \subset C_{G}(<X>) [/tex]. Then, just as above we have [tex] <X> \subset C_{G}(<X>) [/tex] and so <X> is abelian as desired.

i didn't understand how the book went from saying [tex] <X> \subset C_{G}(X) [/tex] and concluding that [tex] X \subset C_{G}(<X>) [/tex]. it seems to me that [tex] X \subset C_{G}(<X>) [/tex] is clear from the hypothesis and i don't see why it was even needed to show that [tex] <X> \subset C_{G}(X) [/tex].

am i missing something here?

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