Non-abelian subgroup of size 6 in A_6

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In summary, the conversation discusses finding a non-abelian subgroup of size 6 in A_6. The individual has tried using permutations in D_6, but encountered a problem with the reflection elements. They receive a hint to find a copy of S_3 in A_6 by interchanging elements two at a time. They are able to find the subgroup {e, (23)(56), (13)(46), (12)(45), (123)(456), (132)(465)} and now need to justify their answer.
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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.
 
  • #3
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.
 

Related to Non-abelian subgroup of size 6 in A_6

1. What is a non-abelian subgroup of size 6 in A6?

A non-abelian subgroup of size 6 in A6 is a subgroup of the alternating group A6 that has 6 elements and does not follow the commutative property. This means that the order in which the elements are multiplied matters.

2. How do you identify a non-abelian subgroup of size 6 in A6?

To identify a non-abelian subgroup of size 6 in A6, you need to find a group of 6 elements that do not follow the commutative property. One way to do this is by finding a subgroup that contains elements with different orders.

3. Is a non-abelian subgroup of size 6 in A6 a normal subgroup?

No, a non-abelian subgroup of size 6 in A6 is not always a normal subgroup. A normal subgroup is a subgroup that remains unchanged under conjugation by any element in the group. A non-abelian subgroup of size 6 in A6 may not fulfill this criterion.

4. What are the possible orders of elements in a non-abelian subgroup of size 6 in A6?

The possible orders of elements in a non-abelian subgroup of size 6 in A6 are 2, 3, and 6. This is because the orders of elements in a subgroup must divide the order of the subgroup, which in this case is 6.

5. Can a non-abelian subgroup of size 6 in A6 be isomorphic to another group?

Yes, a non-abelian subgroup of size 6 in A6 can be isomorphic to another group. Isomorphic groups have the same structure and follow the same rules, even if their elements may be different. Therefore, a non-abelian subgroup of size 6 in A6 can be isomorphic to another group with the same number of elements and non-commutative operation.

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