# Properties of Cosets

__Theorem 1__**:** If , then the right (or left) coset or of is identical to , and conversely.

**Proof:** Let be an arbitrary element of so that .

Again, since is a subgroup, we have

Thus every element of is also an element of .

Hence

Again

This shows that every element of is also an element of .

Hence

From (i) and (ii) it follows that

Similarly, we can show that

Conversely, and similarly .

__Theorem 2__**:** Any two right (or left) cosets of are either disjoint or identical.

**Proof:** Let be a subgroup of a group and let and be two left cosets. Suppose these cosets are not disjoint. Then they possess an element, say , in common. Then may be written as , and also as , where and are in .

Therefore,

Since is a subgroup, .

Let then . Hence

Therefore the two left cosets are identical if they are not disjoint.

Thus either or

A similar result can be shown to hold for right cosets.

__Theorem 3__**:** If is finite, the number of elements in a right (or left) coset of is equal to the order of .

**Proof:** The mapping , defined by , is obviously onto.

It is one-one, since

From the right cancellation law it is onto, since an element belonging to is the image of belonging to .

It follows that the number of elements in a left coset of is the same as that of .

Similarly the number of elements inĀ a left coset of is the same as that of .