Groups: Subgroups

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 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

 

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