Group Theory - specific non-abelian case

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Homework Help Overview

The discussion revolves around finding a specific non-Abelian group where a set defined by the cubes of its elements does not form a subgroup. The subject area is group theory, particularly focusing on properties of non-Abelian groups.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants explore various non-Abelian groups, including the quaternion group and matrix groups, while questioning the properties of the set H defined in the problem. There is also discussion about the permutation group S_3 and its elements' cubes.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts and attempts at identifying suitable groups. Some guidance has been offered regarding the properties of the groups being considered, but no consensus has been reached yet.

Contextual Notes

Participants mention spending considerable time on the problem and express a need for further clarification on the definition and properties of the set H in relation to the groups being examined.

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Homework Statement



Let G be an Abelian group and let H+{x^3 : x is an element of G}

Find a non-Abelian group in which H is not a subgroup

Homework Equations



I wish it was that easy...

The Attempt at a Solution



I looked at the quaternion group, and some other matrix groups, but no luck so far...
 
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What is the first non-abelian group in any reasonable ordering?
 
The permutation group S_3.

I looked at this also... but the cubes of the individual elements seemed closed under the operation of composition, and the identity is present... Four of us have spent about 3 hours on this without much luck, any nudging is greatly appreciated!
 
You should triple-check your work, because matt is never wrong. If G=S_3, what are you thinking H is?
 
Thanks for the help guys! I'm sure glad I don't have to take exams at midnight. A fresh set of eyes in the morning and your confirmation sure helped!
 

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