**What is a subgroup?**

If H is a subset of group G with the restricted operation • from G, then H is a subgroup of G.

The **trivial subgroup** of any group is the subgroup {*e*} consisting of just the identity element.

A **proper subgroup** of a group *G* is a subgroup *H *which is a proper subset of *G, *i.e. *H ≠ G**.*

**Usual Subgroup Criterion**

Let (G,•) be a group and let H be a subset of G. H is a subgroup if and only if:

- The identity is in H

e ∈ H

- H is closed under the operation •

for a,b ∈ H, also a•b ∈ H

- H is closed under inverses

for a ∈ H, a^−1 ∈ H

**Super-efficient Subgroup Criterion**

Let (G,•) be a group and let H be a subset of G. Then H is a subgroup of G if and only if:

- The group is not empty;
- Given a,b ∈ H, then also a•b^-1 ∈ H

This is called the ‘super-efficient’ subgroup criterion as it has less steps than the previous criterion, and therefore these conditions are faster to check. However, this is only true “*when we are dealing with very general situations; if you’re working in an example, to work out a•b^-1 you usually have to work out b^-1 and can already see if it is in the subgroup given. So in examples, it is mostly more sensible to use the usual subgroup criterion.*“

There are proofs for these criterion. Let me know if you want me to go over them in a future post! M x