How many subgroups of size 5 in A_6 have cyclic elements?

  • Thread starter alex07966
  • Start date
Therefore, the total number of subgroups of size 5 in A_6 is 36.In summary, the conversation discusses finding the number of subgroups of size 5 in A_6, which turns out to be 36 due to the fact that every subgroup of order 5 contains four elements and no element is contained in two different subgroups.
  • #1
alex07966
7
0
I need to find the number of subgroups of size 5 in A_6.


I have started by noting that as the subgroup size is 5, a prime, the subgroups must be cyclic. I have worked out that there are 144 elements of order 5 in A_6, but this can't be equal to the number of subgroups (i found two subgroups which have the same elements in!). Someone please help!
 
Physics news on Phys.org
  • #2
If two subgroups of order 5 intersect, can you describe the intersection?
 
  • #3
Ok, I have noticed that <a> = <a^2> = <a^3> = <a^4> for all a in A_6 where a is a 5-cycle. So this means that the number of elements of order 5 must be divided by 4. Hence the answer is 144/4 = 36 subgroups of size 5 in A_6.
Is this correct?
 
  • #4
alex07966 said:
Ok, I have noticed that <a> = <a^2> = <a^3> = <a^4> for all a in A_6 where a is a 5-cycle. So this means that the number of elements of order 5 must be divided by 4. Hence the answer is 144/4 = 36 subgroups of size 5 in A_6.
Is this correct?

Yes, every subgroup of order 5 contains four of them and no element of order 5 is contained in two different subgroups.
 

FAQ: How many subgroups of size 5 in A_6 have cyclic elements?

Question 1: What is A6?

A6 is a subgroup of the symmetric group S6, which consists of all possible permutations of 6 objects. It is also known as the alternating group, as it contains all even permutations.

Question 2: How many subgroups of size 5 are there in A6?

There are a total of 120 subgroups of size 5 in A6. This can be calculated using the formula n!/k!(n-k)!, where n is the total number of elements in the group (6 in this case) and k is the size of the subgroup (5 in this case).

Question 3: What are the elements of a subgroup of size 5 in A6?

The elements of a subgroup of size 5 in A6 are permutations of 5 objects. These permutations can be written in cycle notation, such as (1 2 3 4 5) or (1 2)(3 4 5), where the numbers represent the objects being permuted.

Question 4: How can I determine if a subgroup of size 5 in A6 is a normal subgroup?

A subgroup of size 5 in A6 is a normal subgroup if it is closed under conjugation, meaning that for any element g in the subgroup and any element x in A6, the conjugate gxg-1 is also in the subgroup. This can be checked by calculating all possible conjugates of the elements in the subgroup.

Question 5: How are subgroups of size 5 in A6 related to the group A6 itself?

Subgroups of size 5 in A6 are important in understanding the structure of the group A6. They can be used to find normal subgroups, which in turn can be used to find quotient groups. Additionally, the number of subgroups of size 5 in A6 is a factor in determining the simplicity of the group A6.

Similar threads

Characterizing subgroups of a cyclic group
Replies
9
Views
1K
Cyclic group has 3 subgroups, what is the order of G
Replies
5
Views
2K
Non-abelian subgroup of size 6 in A_6
Replies
2
Views
2K
Verifying a Proof about Maximal Subgroups of Cyclic Groups
Replies
1
Views
2K
Groups that cannot be the direct product of subgroups
Replies
1
Views
2K
Subnormal p-Sylow Subgroup of Finite Group
Replies
9
Views
2K
Z, subgroups and isomorphisms re-attempt
Replies
17
Views
3K
Proof Group Homework: Cyclic if Has Order m & n Elements
Replies
3
Views
1K
Proof that a certain group G contains a cyclic subgroup of order rs
Replies
17
Views
7K
Element Orders For Group Of Order 30
Replies
4
Views
6K

Hot Threads

Recent Insights

Back
Top