Is Every Group with Squared Elements Equal to the Identity Element Abelian?

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If every element in a group G squares to the identity element, then G is proven to be Abelian. The discussion highlights the realization that each element is its own inverse, leading to the conclusion that the product of any two elements ab equals ba. Participants emphasize the learning process, noting that initial confusion can lead to clarity. The exchange illustrates the collaborative nature of problem-solving in group theory. Understanding these properties is crucial for grasping the structure of Abelian groups.
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[SOLVED] An Abelian Group Problem

Homework Statement
Prove: If G is a group where the square of every element equals the identity element, then G is Abelian.

The attempt at a solution
I've been able to prove is that a-1 = a and that (ab)-1 = ba where a and b are in G. Everything else I've done leads into a dead-end. Any tips?
 
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Okay, you know that a-1= a for every element of the group (including ab) and you know that (ab)-1= ba. What do those two together tell you?
 
I see now! Boy do I feel stupid. Thanks.
 
Hey, feeling stupid is the beginning of learning!
 

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