Theorem 1: If , then the right (or left) cosets or of is identical with , 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 .
This shows that every element of is also an element of .
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 .
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 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 belongs to .
It follows that the number of elements in a left coset of is the same as that is .
Similarly the number of elements in a left coset of is the same as that in .