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These theorems seem to be identical but for some reason, the requirement that a group is finite AND abelian is sometimes stated instead of just finite. Could someone let me know if there is a difference in these theorems?

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Cauchy's theorem is for all finite groups not just for abelians. The abelian case is somehow easier to be proved. All in all, if something is true for all groups then it is true for abelian groups whatsoever.

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Cauchy's Theorem is for all finite groups. The proof is sometimes done separately for the abelian case and the non-abelian case. However this is not necessary, see here:

Cauchy's Theorem for Abelian Groups

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  • $\begingroup$ It's just easier to prove the latter, especially in introduction to algebra texts. $\endgroup$ – Don Thousand Mar 23 at 20:05

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