Subgroups

Definition: Submonoid, subgroup

Let be sets, and . If and are Monoids Monoids under the same operation, is called a submonoid of . If and are Groups Groups under the same operation, is called a subgroup of and we write .

We see that every group contains two trivial subgroups: itself and the trivial group . If a subgroup is not one of these two trivial subgroups, we often call it nontrivial.

From this definition the following theorems follow directly and so we state them without proofs.

Theorem

Let be a monoid with unit , and . Then is a submonoid of if and only if and for all .

Theorem

Let be a group and . Then is a subgroup of if and only if is a submonoid of and for all .

We also find that we can find an often easier equivalent requirement for a subset to be a subgroup.

Theorem

Let be a group and where . Then is a subgroup of if and only if , or in other words, for all .

Proof:
If is a subgroup of , it is clear that since is a group and thus closed under composition and inverses. If instead , we can take any and know that , so . This also means for all , which finally means that for all . So is indeed a subgroup of .