QUICK FACTS
Created Jan 0001
Status Verified Sarcastic
Type Existential Dread
Keywords
mathematics, abstract algebra, group, subgroup, generated, commutators, smallest, normal subgroup, quotient group, abelian

Commutator Subgroup

“In the realm of mathematics, and more precisely within the discipline of abstract algebra, one encounters the notion of the commutator subgroup, also known as...”

Contents
  • 1. Overview
  • 2. Etymology
  • 3. Cultural Impact

Definition

In the realm of mathematics , and more precisely within the discipline of abstract algebra , one encounters the notion of the commutator subgroup, also known as the derived subgroup, of a group . This subgroup is defined as the generated subgroup that consists of all possible commutators formed by elements of the group. [1] [2]

The importance of the commutator subgroup stems from the fact that it constitutes the smallest normal subgroup with the property that the quotient group obtained by factoring the original group by this subgroup yields an abelian group. In symbolic terms, if (N) denotes the commutator subgroup of (G), then (G/N) is abelian, and any normal subgroup possessing this characteristic must contain (N). The universal property of the commutator subgroup can be phrased as follows: among all normal subgroups (N) for which the quotient (G/N) is abelian, the commutator subgroup is the unique smallest one; equivalently, it is the intersection of all normal subgroups with abelian quotient. This characterisation makes the commutator subgroup a fundamental invariant that measures how far a group deviates from being abelian.

Commutators

Main article: Commutator

For elements (g) and (h) of a group (G), the commutator of (g) and (h) is conventionally defined as

[ [g,h]=g^{-1}h^{-1}gh . ]

The commutator ([g,h]) equals the identity element (e) if and only if (gh=hg), i.e. precisely when (g) and (h) commute. In general one has the relation

[ gh = hg,[g,h] . ]

However, the notation is somewhat arbitrary and there is a non‑equivalent variant definition for the commutator that places the inverses on the right‑hand side of the equation:

[ [g,h]=ghg^{-1}h^{-1}, ]

in which case (gh\neq hg,[g,h]) but instead (gh=[g,h],hg).

An element of (G) of the form ([g,h]) for some (g) and (h) is called a commutator. The identity element (e=[e,e]) is always a commutator, and it is the only commutator if and only if (G) is abelian.

Here are some simple but useful commutator identities, true for any elements (s,,g,,h) of a group (G):

  • ([g,h]^{-1}=[h,g]),
  • ([g,h]^{s}=[g^{s},h^{s}]),

where (g^{s}=s^{-1}gs) (or, respectively, (g^{s}=sgs^{-1})) is the conjugate of (g) by (s),

The first and second identities imply that the set of commutators in (G) is closed under inversion and conjugation. If in the third identity we take (H=G), we get that the set of commutators is stable under any endomorphism of (G). This is in fact a generalization of the second identity, since we can take (f) to be the conjugation automorphism on (G), (x\mapsto x^{s}), to obtain the second identity.

Nevertheless, the product of two or more commutators need not be a commutator. A generic example is ([a,b][c,d]) in the free group on (a,b,c,d). It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96; in fact there are two non‑isomorphic groups of order 96 with this property. [3]

Derived series

This construction can be iterated:

[ G^{(n)} := [,G^{(n-1)},,G^{(n-1)},]\qquad n\in\mathbf{N}. ]

The groups (G^{(2)},G^{(3)},\dots) are called the second derived subgroup, third derived subgroup, and so forth, and the descending chain

[ G^{(2)}\triangleleft G^{(1)}\triangleleft G^{(0)}=G ]

is called the derived series. This series should not be confused with the lower central series , whose terms are defined recursively by

[ G_{n}:=[,G_{n-1},,G,], ]

starting with (G_{0}=G).

For a finite group the derived series eventually stabilises in a perfect group , which may or may not be trivial. For an infinite group the derived series need not terminate at a finite stage; one can extend it transfinitely by using ordinal numbers and transfinite recursion . In this manner the process culminates in the perfect core of the group, the largest perfect subgroup that remains after all derived‑series steps have been performed.

Abelianization

Given a group (G), a quotient group (G/N) is abelian if and only if the derived subgroup ([G,G]) is contained in (N). The quotient (G/[G,G]) is an abelian group called the abelianization of (G) or “(G) made abelian”. [4] It is usually denoted by (G^{\operatorname{ab}}) or (G_{\operatorname{ab}}).

There is a useful categorical interpretation of the canonical homomorphism

[ \varphi : G \longrightarrow G^{\operatorname{ab}} . ]

Namely (\varphi) is universal for homomorphisms from (G) to an abelian group (H): for any abelian group (H) and any group homomorphism (f : G \to H) there exists a unique homomorphism (F : G^{\operatorname{ab}} \to H) such that (f = F \circ \varphi). As is typical for objects characterised by universal mapping properties, this universal property guarantees the uniqueness of the abelianization (G^{\operatorname{ab}}) up to a canonical isomorphism, while the explicit construction (G \to G/[G,G]) guarantees its existence.

The abelianization functor is the left adjoint of the inclusion functor from the category of abelian groups to the category of groups. The existence of the abelianization functor (\mathbf{Grp} \to \mathbf{Ab}) makes (\mathbf{Ab}) a reflective subcategory of the category of groups, defined as a full subcategory whose inclusion functor possesses a left adjoint.

Another important interpretation of (G^{\operatorname{ab}}) is as the first homology group (H_{1}(G,\mathbb{Z})) of (G) with integral coefficients; this identification links group theory with algebraic topology.

Classes of groups

A group (G) is an abelian group if and only if its derived subgroup is trivial, i.e. ([G,G]={e}). Equivalently, this holds precisely when (G) coincides with its own abelianization. See the discussion above for the definition of a group’s abelianization.

A group (G) is a perfect group if and only if its derived subgroup equals the group itself, ([G,G]=G). Equivalently, a perfect group is characterised by a trivial abelianization. This situation can be viewed as the “opposite” of an abelian group.

A group for which (G^{(n)}={e}) for some natural number (n) is called a solvable group ; this class strictly contains the abelian groups, which correspond to the case (n=1).

A group for which (G^{(n)}\neq{e}) for every natural number (n) is called a non‑solvable group.

A group possessing a transfinite ordinal (\alpha) such that (G^{(\alpha)}={e}) is called a hypoabelian group ; this notion weakens the notion of solvability, which corresponds to the finite‑stage case.

Perfect group

• Main article: Perfect group

Whenever a group (G) satisfies ([G,G]=G), it is termed a perfect group. Perfect groups include all non‑abelian simple groups and many classical families of linear groups, notably the special linear groups (\operatorname{SL}_{n}(k)) over a field (k) (with the usual exceptions noted in the literature).

Examples

These examples illustrate how the derived subgroup can be trivial, finite, or even the whole group, depending on the ambient structure.

Map from Out

Since the derived subgroup is characteristic , any automorphism of (G) induces an automorphism of the abelianization. Because the abelianization is itself abelian, inner automorphisms act trivially on it. Consequently one obtains a natural homomorphism

[ \operatorname{Out}(G) \longrightarrow \operatorname{Aut}(G^{\operatorname{ab}}), ]

which plays a central role in the study of the outer automorphism group and its interaction with the abelian structure of (G).

See also

Notes

  • ^ Dummit & Foote (2004)
  • ^ Lang (2002)
  • ^ SuĂĄrez‑Alvarez
  • ^ Fraleigh (1976, p. 108)
  • ^ Suprunenko, D.A. (1976), Matrix groups, Translations of Mathematical Monographs, American Mathematical Society, Theorem II.9.4