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## Main Question or Discussion Point

This is not homework. Self-study. And I'm really enjoying it. But, as I'm going through this book ("A Book of Abstract Algebra" by Charles C. Pinter) every so often I run into a problem or concept I don't understand.

Let G be a finite abelian group, say G = (e,a1, a2, a3,...,an).

Prove that (a1*a2*...an)^2 = e.

So, it has a finite number of elements and it's a group. So it's associative, has an identity element and an inverse as elements of G, and as it's abelian so it's also commutative. But I don't see how squaring the product of its elements leads to the identity element e.

Wait. Writing this has me thinking that each element might be being 'multiplied' by it's inverse yielding e for every pair, which when all mutiplied together still yields e, even when ultimately squared. Could that be the answer, even though I may not have stated it elegantly? There's no one I can ask so I brought it to this forum.

Let G be a finite abelian group, say G = (e,a1, a2, a3,...,an).

Prove that (a1*a2*...an)^2 = e.

So, it has a finite number of elements and it's a group. So it's associative, has an identity element and an inverse as elements of G, and as it's abelian so it's also commutative. But I don't see how squaring the product of its elements leads to the identity element e.

Wait. Writing this has me thinking that each element might be being 'multiplied' by it's inverse yielding e for every pair, which when all mutiplied together still yields e, even when ultimately squared. Could that be the answer, even though I may not have stated it elegantly? There's no one I can ask so I brought it to this forum.