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

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Homework Help Overview

The discussion revolves around a group theory problem concerning the properties of a group where the square of every element equals the identity element. The subject area is abstract algebra, specifically focusing on group properties and Abelian groups.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to prove that the group is Abelian by showing that the inverse of each element is the element itself and exploring the implications of the relationship between the product of two elements and their inverses.

Discussion Status

Participants are engaging in a productive dialogue, with one offering insights that connect the properties of the group elements. There is an acknowledgment of the learning process, with encouragement provided to the original poster.

Contextual Notes

There is an implicit assumption that the group in question adheres to the properties of a mathematical group, and the discussion is framed within the context of proving a specific property related to group operations.

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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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