abstract algebra groups

[perusal]

I think I am beginning to get my head around groups. However, I am struggling with the notion of subgroups and cosets. I understand the proofs but am having difficulty thinking of an actual example of a subgroup coset.

Would someone be kind enough to provide an example?

 

[shoumeiowari]

The set of integers is a subgroup of the set of real numbers under addition.

 

[pka]

I think I am beginning to get my head around groups. However, I am struggling with the notion of subgroups and cosets. I understand the proofs but am having difficulty thinking of an actual example of a subgroup coset.
Would someone be kind enough to provide an example?

If we have any group \(\displaystyle \mathcal{G}\) and a subset \(\displaystyle \mathcal{H}\subseteq\mathcal{G}\) that has the property that whenever \(\displaystyle \{a,b\}\subset\mathcal{H}\) it necessarily followers that \(\displaystyle ab^{-1}\in\mathcal{H}\) then \(\displaystyle \mathcal{H}\) is a subgroup of \(\displaystyle \mathcal{G}\).

 

[HallsofIvy]

Given a subgroup, H, of a group, G, written Ha for a a member of G, is the set of all products of a with every member of H. For example, suppose G is the group of integers modulo 4. Then the members of the group are {0, 1, 2, 3} with operation
0+ 0= 0, 0+ 1= 1, 0+ 2= 2, 0+ 3= 3
1+ 0= 1, 1+ 1= 2, 1+ 2= 3, 1+ 3= 0
2+ 0= 2, 2+ 1= 3, 2+ 2= 0, 2+ 3= 1
3+ 0= 3, 3+ 1= 0, 3+ 2= 1, 3+ 3= 2,

H= {0, 2} is a subgroup:
0+ 0= 0, 0+ 2= 2, 2+ 0= 2, 2+ 2= 0

Its cosets are
H0= {0+ 0, 2+ 0, 3+ 0}= {0, 2, 3}
H1= {0+ 1, 2+ 1, 3+ 1}= {1, 3, 0}
H2= {0+ 2, 2+ 2, 3+ 2}= {2, 0, 1}= H0
H3= {0+ 3, 2+ 3, 2+ 2}= {3, 1, 0}= H1

 

[perusal]

Thanks everyone. Huge help, pretty confident I understand now.