# Elements in only 1 left/right coset

• Wiemster
In summary, the conversation discusses the concept of right cosets of a subgroup in a group and how every element in the group belongs to exactly one right coset. The theorem is proved using the concept of equivalence relations and it is noted that cosets are either equal or disjoint. An example of a generalized torus is given, but it is clarified that (R/Z)^n is not isomorphic to the unit circle in C.
Wiemster
My lecture notes state that every element of a group belongs to exactly one right and one left coset of a certain subgroup, but I don't see why this should be the case, so I tries to prove it:

with S and S' elements of the subgroup H of G we then have that if a certain element X of the group G belongs to the coset TH but to T'H as well

i.e. X=TS=T'S' we should have that T=T' right?

This means TS(S'^-1)=T'E with S(S'^-1) again an element of H, but I don't see why T should be T', or did I misintepreted the theorem?

(PS: Somewhat further they use as an example a generelaized torus: Rn / Zn = (R/Z)n which is supposed to b homomorphic to the unit circle. But I don't really see what this 'factor group' represents... )

The right cosets of H in G are the equivalences classes of the equivalence relation ~ on G given by x~y iff xy^-1 in H. They are equivalence classes, so they partition G into a family of mutually disjoint nonempty subsets. So every element in G belongs to exactly 1 right coset(to exactly 1 equivalence class). So if x belongs to two right cosets say Ha and Hb, then it follows that Ha = Hb.

I just said this in words, didn't prove anything. If you want to prove it, you can,

1) Let R be an equivalence relation on a set S. Show every element of S is in exactly one equivalence class.

2) Show the relation defined, ~, is an equivalence relation on a group G(with H being a subgroup).

3) Show the equivalence classes of ~ are the right cosets of H in G.

Then it follows immediately.

Cosets are either equal or disjoint. It is a natural consequence of the definition of a group.

(R/Z) is isomorphic to the unit circle in C. (R/Z)^n is not.

## What does it mean for an element to be in only 1 left coset?

For an element to be in only 1 left coset means that it is only in one of the subsets formed by the left coset partition of a group. This means that the element is not repeated in any other left coset subset.

## What is the significance of an element being in only 1 right coset?

An element being in only 1 right coset means that it is only in one of the subsets formed by the right coset partition of a group. This indicates that the element is not repeated in any other right coset subset.

## Can an element be in both a left coset and a right coset?

Yes, it is possible for an element to be in both a left coset and a right coset. However, it must be in different subsets within each coset partition. This means that it cannot be in the same subset in both the left and right coset partitions.

## How do elements in only 1 left/right coset affect the overall structure of a group?

Elements in only 1 left/right coset do not affect the overall structure of a group significantly. They simply indicate that the element is not repeated in any other left/right coset subset, which can provide insight into the structure and properties of the group.

## What is the relationship between elements in only 1 left coset and elements in only 1 right coset?

There is no specific relationship between elements in only 1 left coset and elements in only 1 right coset. They are simply two different ways of partitioning a group and indicating that the element is not repeated in any other subset within that particular partition.

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