Subgroup

In group theory, a subgroup of a group is a subset which is itself a group with respect to the same operations.

Formally, a subset S of a group G is a subgroup if it satisfies the following conditions:

• The identity element of G is an element of S;
• S is closed under taking inverses, that is, $x\in S\Rightarrow x^{-1}\in S$;
• S is closed under the group operation, that is, $x,y\in S\Rightarrow xy\in S$.

These correspond to the conditions on a group, with the exception that the associative property is necessarily inherited.

It is possible to replace these by the single closure property that S is non-empty and $x,y\in S\Rightarrow xy^{-1}\in S$.

The group itself and the set consisting of the identity element are always subgroups.

Particular classes of subgroups include:

• Characteristic subgroup [r]: A subgroup which is mapped to itself by any automorphism of the whole group. ^[e]
• Essential subgroup [r]: A subgroup of a group which has non-trivial intersection with every other non-trivial subgroup. ^[e]
• Normal subgroup [r]: Subgroup N of a group G where every expression g-1ng is in N for every g in G and every n in N. ^[e]

Specific subgroups of a given group include:

• Centre of a group [r]: The subgroup of a group consisting of all elements which commute with every element of the group. ^[e]
• Commutator subgroup [r]: The subgroup of a group generated by all commutators. ^[e]
• Frattini subgroup [r]: The intersection of all maximal subgroups of a group. ^[e]

Cosets

The left cosets of a subgroup H of a group G are the subsets of G of the form x H for a particular element x of G:

$xH=\{xh:h\in H\}.\,$

The right cosets H x are defined similarly:

$Hx=\{hx:h\in H\}.\,$

The left cosets partition the group G, any two cosets $xH$ and $yH$ are either equal or disjoint. This may be proved directly, or deduced from the observation that the left cosets are the equivalence classes for the equivalence relation ${\stackrel {H}{\sim }}$ defined by

$x{\stackrel {H}{\sim }}y\Leftrightarrow x^{-1}y\in H.\,$

Similar remarks apply to the right cosets. A subgroup is normal if the left cosets agree with the right cosets for all elements.

References

• Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 7-8.