Subgroups

A subgroup is simply a subset of a gropup which is also a group. Since a subgroup necessarily contains the identity and its own inverses, and the associative law is already known for all possible triple products including those which might come from a prospective subgroup, the only doubt of consequence in verifying a subgroup lies in checking closure. That would be the normal sequence, locating the identity, finding inverses and examining products; but sometimes a proof can be shortened by simply applying the criterion that a subset is a subgroup if it contains all quotients.

The reasoning is that if $x=y$ in $xy^{-1}$, it will turn up $xx^{-1}=e$, giving a check for the identity. Once $e$ is there, putting $e$ and $x$ in the test runs through the inverses because $ex^{-1} = x^{-1}$. Finally pairing $x$ and $y^{-1}$ locates all the products due to $x(y^{-1})^{-1} = xy$, which was what was intended from the beginning. So this tricky wording allows a three-stage procedure to be described by one single algorithm. Note that it is not very practical when applied to a group table because it won't work without knowing what the inverses are. But later on, when complicated groups have been constructed with the help of algebraic formulas, including an extra inverse in a calculation could be preferable to checking all the special cases.

Subgroups are ordered by inclusion, so that the full group is the maximum subgroup, the unit class of the identity the minimum subgroup. The intersection of two subgroups is their lower bound, but the least upper bound is a more complicated matter; it is usually larger than their union. It also has to contain all their finite products, but that is actually sufficient; the resulting assemblage is called a hullhull, and can even be formed for subsets which are not subgroups. The smallest subgroup containing a given element, or specified set of elements, is often useful, and contains at least their powers and inverses.