Cosets and Lagrange’s Theorem

Cosets

Coset

Let be a group, a subgroup and . The following subsets are called cosets of in :
We denote as the space of left cosets.

Proposition

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

Proof If , then we are allowed to pick some element , so that . Then , so , and similarly , thus . Moreover, any element belongs to , so .

Lagrange’s Theorem

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

Proof Let be the distinct cosets. Then .

Proposition

Let be a finite group and . Then . It follows that for all . In particular, if there exists such that then is cyclic.

Proof Consider . By Lagrange’s Theorem, we have . Since is cyclic, , which implies that . Moreover, if , then will have order , which means .

Corollary

A group of prime order is cyclic.

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

Proposition

If and are subsets of a finite group , and , then is generated by or .

Proof Denote the subgroup generated by as and the subgroup generated by as . Then and . By Lagrange’s Theorem, we have and . If neither nor is equal to , then and , which implies that , contradicting the assumption. Therefore, either or .

Index

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 .

Remark

The “in particular” part is a direct consequence of Lagrange’s theorem.

Proposition

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

Proof Suppose such that is the smallest prime factor of . Consider the standard action of on the left coset space by left multiplication. This gives a homomorphism The kernel of this homomorphism is the normal core of , which is a normal subgroup of . The image of is a subgroup of the symmetric group , so its order divides . By the First Isomorphism Theorem, we have In particular, divides . So divides (by definition of gcd). It follows that , and hence . Therefore, is normal in by the theorem.