Subgroup

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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, ${\displaystyle x\in S\Rightarrow x^{-1}\in S}$;
• S is closed under the group operation, that is, ${\displaystyle x,y\in S\Rightarrow xy\in S}$.

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 ${\displaystyle x,y\in S\Rightarrow xy^{-1}\in S}$.

Examples[edit]

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]

Properties[edit]

The intersection of any family of subgroups is again a subgroup. We can therefore define the subgroup generated by a subset S of a group G, denoted ${\displaystyle \langle S\rangle }$, to be the intersection of all subgroups of G containing S. The union of two subgroups is not in general a subgroup (indeed, it is only a subgroup if one component of the union contains the other). Instead, we may define the join of two subgroups to the subgroup generated by their union.

Cosets[edit]

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:

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

The right cosets H x are defined similarly:

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

The subgroup H is itself one of its own cosets, namely that on the identity element.

The left cosets partition the group G, any two cosets ${\displaystyle xH}$ and ${\displaystyle yH}$ 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 ${\displaystyle {\stackrel {H}{\sim }}}$ defined by

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

Similar remarks apply to the right cosets. In general the two partitions of the group defined by the left cosets and by the right cosets are not the same. A subgroup is normal if and only if the left cosets agree with the right cosets for all elements.

Index[edit]

The index of a subgroup H of a group G, denoted ${\displaystyle [G:H]}$ is the number (if finite) of cosets of H in G. Two cosets may be put into one-to-one correspondence ${\displaystyle xH\leftrightarrow yH}$ by ${\displaystyle xh\leftrightarrow yh}$, so if the cosets are finite then they all have the same order. We can now deduce

Lagrange's Theorem: In a finite group the order of a subgroup multiplied by its index equals the order of the group:

${\displaystyle \vert G\vert =\vert H\vert \cdot [g:h].\,}$

In particular the order of a subgroup divides the order of the group, and the order of an element divides the orderof the group.

Maximal subgroup[edit]

A subgroup M of G is maximal if M is not the whole of G but there is no other subgroup H strictly between M and G.

References[edit]

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