The Unapologetic Mathematician

Mathematics for the interested outsider

Subgroups

A subgroup is pretty straightforward. It’s a little group living inside a bigger group. If you’ve got a group G and some collection H of elements of G so that H is a group using the same composition as G, then H is a subgroup. To be more explicit, you need that

  • If x and y are in H then xy is in H.
  • If x is in H then x^{-1} is in H.
  • The identity e of G is in H. [added at the suggestion of Toby Bartels]

We say that a subgroup is “closed” under composition and inverse, meaning that if we start with elements of H and take compositions and inverses we never leave the subgroup.

The collection of all even integers is a subgroup of the group {\mathbb Z} of all integers (with addition as the operation).
The subset \{e,(1\,2\,3),(1\,3\,2)\} is a subgroup of the group S_3.
Every group has two “trivial” subgroups: the whole group itself, and the subgroup consisting of just the identity element.
There are two ways of getting subgroups that I want to spend a bit more time on: “images” and “kernels”.
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February 13, 2007 Posted by | Algebra, Group theory, Subgroups and Quotient Groups | 7 Comments