# Tour:Sufficiency of subgroup criterion

From Groupprops

**This article adapts material from the main article:** sufficiency of subgroup criterion

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Subsemigroup of finite group is subgroup|UP: Introduction two (beginners)|NEXT: Manipulating equations in groupsExpected time for this page: 8 minutes

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WHAT YOU NEED TO DO:

- Recall the various definitions of subgroup, including the subgroup criterion
- Try proving that the subgroup criterion is necessary and sufficient
- Read below the proof of sufficiency
- Compare with the result we just saw for finite groups (any nonempty multiplicatively closed subset is a subgroup)

PONDER:

- What axioms of group structure play a role in this proof?
- In what ways are things different for finite groups, and why?

## Statement

For a subset of a group , the following are equivalent:

- is a subgroup, viz is closed under the binary operation of multiplication, the inverse map, and contains the identity element
- is a nonempty set closed under left quotient of elements (that is, for any in , is also in )
- is a nonempty set closed under right quotient of elements (that is, for any in , is also in )

## Proof

We shall here prove the equivalence of the first two conditions. Equivalence of the first and third conditions follows by analogous reasoning.

### (1) implies (2)

Clearly, if is a subgroup:

- is nonempty since contains the identity element
- Whenever are in so is and hence

### (2) implies (1)

Suppose is a nonempty subset closed under left quotient of elements. Then, pick an element from . (**VIDEO WARNING**: In the embeddded video, the letter is used in place of , which is a little unwise, but the spirit of reasoning is the same).

- is in : Set to get is contained in , hence is in
- : Now that is in , set to get is also in , so is in
- : Set . The previous step tells us both are in . So is in , which tells us that is in .

Thus, satisfies all the three conditions to be a subgroup.

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Subsemigroup of finite group is subgroup|UP: Introduction two (beginners)|NEXT: Manipulating equations in groups