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 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.