Cosets and Lagrange’s Theorem

Coset

Let be a group, a subgroup and . The following subsets are called cosets of in :

Proposition

Let be a subgroup of a group . Then the left (or right) cosets form a partition of . That is and for either or .

Lagrange’s Theorem

Let be a subgroup of a finite group . Then divides .

Proof Let be the distinct cosets.Then .

Corollary

Let be a finite group and . Then . It follows that for all .

Proof Consider . By Lagrange’s Theorem, we have . Since is cyclic, , which implies that .

Index

The index of a subgroup in a group , denoted , is defined as the cardinality of the set of (left) cosets of in . In particular, if is a finite group, then .

Corollary

A group of prime order is cyclic.

Proof Suppose group has prime order. Then for all subgroup , by corollary, for all ,we have , thus or , indicating that is a cyclic group.

Proposition

A subgroup whose index is the smallest prime dividing the order of is normal. In, particular, any subgroup of index 2 in is automatically normal.