Non-abelian subgroup of size 6 in A_6

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SUMMARY

The discussion focuses on identifying a non-abelian subgroup of size 6 within the alternating group A_6. The subgroup in question is isomorphic to the symmetric group S_3, which can be represented by the permutations {e, (23)(56), (13)(46), (12)(45), (123)(456), (132)(465)}. The challenge lies in demonstrating that this subgroup is isomorphic to D_6, the dihedral group of order 6, while ensuring that the reflection elements, which are 2-cycles, do not belong to A_6. The solution involves interchanging elements two at a time to construct the subgroup.

PREREQUISITES
  • Understanding of group theory concepts, specifically non-abelian groups.
  • Familiarity with the structure and properties of the alternating group A_6.
  • Knowledge of dihedral groups, particularly D_6.
  • Experience with permutation notation and cycle decomposition.
NEXT STEPS
  • Study the properties of the alternating group A_6 in detail.
  • Learn about the structure and applications of dihedral groups, focusing on D_6.
  • Explore the concept of isomorphism in group theory, particularly between S_3 and D_6.
  • Practice constructing subgroups within symmetric and alternating groups using permutation techniques.
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in group theory, algebra students, and anyone interested in the properties of symmetric and alternating groups.

alex07966
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I need to find just one non-abelian subgroup of size 6 in A_6.

I have started by noting that the subgroup must be isomorphic to D_6 and then tried to use the permutations in D_6 that sends corners to corners. I then came across the problem that the reflection elements in D_6 consist of 2-cycles and hence are not elements of A_6.
I am now very stuck... please help :)
 
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There's really only one nonabelian group of order 6, S_3. Can you think of a sort of natural way to find a copy of S_3 in A_6? Hint: interchange elements two at a time.
 
Thanks a lot. I have found what the subgroup is now: {e, (23)(56), (13)(46), (12)(45), (123)(456), (132)(465)}. I need to justify my answer so I need to show that this is isomorphic to D_6 and therefore is a subgroup. I'll have a go.
 

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