Finding all of the right cosets of H in G

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In summary, the right cosets of H in G, where G is a cyclic group of order 10 and H is the subgroup generated by a^2, are {a^0, a^2, a^4, a^6, a^8} and {a^1, a^3, a^5, a^7, a^9}. These cosets are disjoint and form a partition of G.
  • #1
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Homework Statement


"Write out all the right cosets of H in G where G = (a) is a cyclic group of order 10 and H = (a^2) is the subgroup of G generated by a^2."


Homework Equations


- If G = (a), then G = {a^i | i=0,-1,1,-2,2...}.
- A right coset is the set Hb = {hb | h is in H}
- Order of G is 10, so G has ten elements. Order of H has to divide that.
- G is the disjoint union of all its right cosets.
- The right cosets of H in G are disjoint.
- The

The Attempt at a Solution


I guess I'm a bit confused, due to the order being ten but not being given any elements to work with.

I reasoned that H = {a^(2i) | i=0,-1,1...}, which led to me thinking that at least one right coset was:
Ha = {a^(3i) | i = 0, -1, 1,...}

But I'm not sure where I should stop with the i's, leading to me being confused as to how many elements would be in that right coset (if that even is one of the right cosets).

Am I even on the right track?
 
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  • #2
Let's be a little more concrete here. If G has order 10 and is generated by a then G={a^0,a^1,...a^9}. Let's call a^0=1. The group identity. So a^10=a^0=1. Now, what is H?
 
  • #3
Then that makes H = G, doesn't it?

H = {a^0, a^3, a^6, a^9, a^2, a^5, a^8, a^1, a^4, a^7}, if I keep adding 3 to the powers and cycling through once they get to ten. Then it's clear that H = G.
 
  • #4
hb1547 said:
Then that makes H = G, doesn't it?

H = {a^0, a^3, a^6, a^9, a^2, a^5, a^8, a^1, a^4, a^7}, if I keep adding 3 to the powers and cycling through once they get to ten. Then it's clear that H = G.

H is generated by a^2. Not a^3.
 
  • #5
Ahh right. So would it be:

H = {a^0, a^2, a^4, a^6, a^8}?

Then that leads me to infer that the other right coset must be:
{a^1, a^3, a^5, a^7, a^9}

But I'm not sure how quite to show that -- is it because the first coset is He, where it's acting on the identity, and the other one is Ha, where it's acting on a?
 
  • #6
hb1547 said:
Ahh right. So would it be:

H = {a^0, a^2, a^4, a^6, a^8}?

Then that leads me to infer that the other right coset must be:
{a^1, a^3, a^5, a^7, a^9}

But I'm not sure how quite to show that -- is it because the first coset is He, where it's acting on the identity, and the other one is Ha, where it's acting on a?

Yes, I think you got it. There are two cosets.
 
  • #7
Okay awesome, thank you very much.

I guess what I'm confused about is that the definition for a cyclic group said,
{a^i | i=0,-1,1,-2,2...}.

Here, we only used the positive values of i, and I guess I'm unclear as to why we did that.
 
  • #8
Because if the group has order 10 then a^(-1)=a^9. Right? You don't need the negative powers and you don't need powers over 9. They are duplicates of the other powers.
 
  • #9
Ahh, so the negative powers cycle through as well. Okay thanks! That helps a lot!
 

1. What are cosets in group theory?

Cosets in group theory are a fundamental concept used to understand the structure of a group. A coset is a subset of a group formed by multiplying a fixed element (known as the "representative" of the coset) by all elements of the group. The collection of all cosets of a subgroup forms a partition of the group.

2. How do you determine the number of cosets in a group?

The number of cosets in a group is equal to the index of the subgroup in the group. The index is the number of distinct left or right cosets, and it can be calculated using Lagrange's theorem, which states that the order of a subgroup must divide the order of the group.

3. How do you find all the right cosets of a subgroup in a group?

To find all the right cosets of a subgroup in a group, you first need to select a representative element for the coset. Then, you multiply this representative by all elements of the subgroup to obtain the other elements of the coset. Repeat this process for all elements in the group to obtain all the right cosets.

4. What is the significance of finding all the right cosets of a subgroup in a group?

Finding all the right cosets of a subgroup in a group allows us to understand the structure of the group and its subgroups. It also helps us to determine the size and properties of the group, such as its order and whether it is a normal subgroup.

5. Can the process of finding all the right cosets be applied to all groups?

Yes, the process of finding all the right cosets can be applied to all groups. However, it may be more challenging for certain groups with complex structures. In some cases, a group may not have any subgroups, which means there are no cosets to be found.

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