Show that if (ab)^2=(a^2)(b^2) for all elements a , b of a group G,then G is abelian
I know how to prove that it is associative, but have no idea how to prove if it has identity and inverse elements.
and I know a group G with the property that a ◦ b = b ◦ a for all a, b ∈ G is called abelian.
how to achieve a ◦ b = b ◦ a ??
Thanks
Abelian group?
回答 (1)
2015-09-22 4:43 am
✔ 最佳答案
You're overthinking it:abab = (ab)^2 = (a^2)(b^2) = aabb
so multiply on the left by a^(-1) and on the right by b^(-1) to get
ba = ab
The question says that G is a group, and you just have to show it's abelian. So, you can use inverses and identities as much as you like. You could presumably cook up a monoid where the analogous statement is false, but I'd have to think about it.
收錄日期: 2021-04-18 00:23:40
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