Proving Coset Properties in Abstract Algebra

In summary, the conversation discusses the properties of cosets and how to prove that aH and bH are equal if and only if a^-1b is an element of the subgroup H. The definition of a left coset with a is also clarified. It is mentioned that cosets partition the group and that for two cosets to be equal, there must exist an element c in aH that equals bh' for some h' in H.
  • #1
vwishndaetr
87
0
Question:

Prove the following properties of cosets.

Given:

Let H be a subgroup and let a and b be elements of G.

[tex] H\leq\ G [/tex]

Statement:

[tex] aH=bH \ if\ and\ only\ if\ a^{-1}b\ \epsilon\ H [/tex]

The statement is what I have to prove.

My issue is I don't know how to start off the problem. When I first looked at the statement. I wanted to say that it is only true when a=b. But there is not talk of the groups being abelian. So what I thought was a start to some thinking, did not take me very far.
 
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  • #2
First of all, you need to say "H is a subgroup of G" or the problem doesn't make sense. Now, what is the definition of "aH" and "bH"? Your remark that "I wanted to say that it is only true when a=b" indicates that you are not clear on that definition.
 
  • #3
[tex]
H\leq\ G
[/tex]

That means H is a subgroup of G. So clearly stated.

a and b are elements of G, and "aH" is a left coset with "a" and "bH" is a left coset with "b."
 
  • #4
vwishndaetr said:
a and b are elements of G, and "aH" is a left coset with "a" and "bH" is a left coset with "b."

But then what is the definition of a "left coset with a"?
 
  • #5
foxjwill said:
But then what is the definition of a "left coset with a"?
[tex]
aH= \{a*h\ |\ h\ \epsilon\ H }\
[/tex]

H is a subset. Where h is an element of the set H.
 
  • #6
vwishndaetr said:
[tex]
aH= \{a*h\ |\ h\ \epsilon\ H }\
[/tex]

H is a subset. Where h is an element of the set H.

Right. So, what does it mean for the set aH to be equal to the set bH?

Oh, and you can type the "in" symbol using a "\in" and the "not in" symbol using "\notin". I think it formats it better that way.
 
  • #7
Remember that aH and bH partition the group (since they are equivalence classes defined by a=b if b = ah for some h in H), if you had any two cosets which had a nonzero intersection, then the transitivity of equivalence classes would automatically make the two equal. So to begin with, you know that if c is in aH, then c equals ah for h in H. if d is in bH, it equals d = bh', for h' in H. Your condition for the equivalence classes being equal is that there exists an element c in aH, such that c=bh' for some h' in H.
 
  • #8
Yup thanks. I had one of my professors explain it to me. Forgot to post up.

Thanks though! :)
 

1. What are cosets in abstract algebra?

Cosets are a fundamental concept in abstract algebra that involves dividing a group into subsets based on the elements of a subgroup. They are formed by multiplying each element of the subgroup by a fixed element from the original group.

2. How are cosets related to normal subgroups?

In order for cosets to be well-defined, the subgroup must be a normal subgroup of the original group. This means that the left and right cosets of the subgroup are equal, and the cosets form a group under the operation of the original group.

3. What is the significance of cosets in group theory?

Cosets play a crucial role in group theory as they allow us to study the structure of a group by breaking it down into smaller, more manageable subgroups. They also help us understand the relationship between subgroups and normal subgroups.

4. How do we determine the number of cosets in a group?

The number of cosets in a group is equal to the index of the subgroup, which is the number of left or right cosets in the group. It can be calculated by dividing the order of the group by the order of the subgroup.

5. Can cosets be used to define a new group?

Yes, cosets can be used to define a new group called the quotient group, which is formed by taking the cosets of a normal subgroup. The elements of the quotient group are the cosets, and the operation is defined by the operation of the original group.

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