Commutator subgroup - Citizendia

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative.

The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.

1 Definition
2 Notes
3 Examples
4 Properties
5 Footnotes
6 See also

Definition

Given a group G the commutator subgroup [G,G] (also called the derived subgroup, and denoted G′ or G⁽¹⁾) of G is the subgroup generated by all the commutators^[1] $[g, h]: = g - 1 h - 1 g h$ of elements of G, that is

$[G,G] = \langle g^{-1}h^{-1}gh \, | \, g, h \in G\rangle .$

The commutator subgroup is a fully characteristic subgroup: it is closed under all endomorphisms of the group. In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the In Mathematics, a Subgroup of a group is fully characteristic (or fully invariant) if it is Invariant under every Endomorphism In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself In particular, the commutator subgroup is a normal subgroup.

Derived Series

This construction can be iterated:

G (0) : = G

$G^{(n)} := [G^{(n-1)},G^{(n-1)}] \quad n \in \mathbb{N}$

The groups $G^{(2)}, G^{(3)}, \ldots$ are called the second derived subgroup, third derived subgroup, and so forth, and the descending normal series

$\cdots \triangleleft G^{(2)} \triangleleft G^{(1)} \triangleleft G^{(0)} = G$

is called the derived series. In Mathematics, a subgroup series is a chain of Subgroups 1 = A_0 \leq A_1 \leq \cdots \leq A_n = G This should not be confused with the lower central series, whose terms are $G n : = [G n - 1, G]$ , not $G (n) : = [G (n - 1), G (n - 1)]$ . In Mathematics, especially in the fields of Group theory and Lie theory, a central series is a kind of Normal series of Subgroups or

For a finite group, the derived series terminates 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, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core of the group. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. Transfinite induction is an extension of Mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals. In Mathematics, in the field of Group theory, the perfect core (or perfect radical) of a group is its largest perfect Subgroup

Special Classes of Groups

A group G is an abelian group if and only if the derived group is trivial: $[G, G] = e$ . An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Equivalently, if and only if the group equals its abelianization.

A group G is a perfect group if and only if the derived group equals the group itself: $[G, G] = G$ . In Mathematics, in the realm of Group theory, a group is said to be perfect if it equals its own Commutator subgroup, or equivalently if the Equivalently, if and only if the abelianization of the group is trivial. This is "opposite" to abelian.

A group with $G (n) = {e}$ for some n in N is called a solvable group; this is weaker than abelian, which is the case $n = 1$ . In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally

A group with $G (α) = {e}$ for some ordinal number, possibly infinite, is called a hypoabelian group; this is weaker than solvable, which is the case $n$ is finite (a natural number). In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. In Mathematics, in the field of Group theory, the perfect core (or perfect radical) of a group is its largest perfect Subgroup

Abelianization

The quotient group $G / [G, G]$ is an abelian group called the abelianization of G or G made abelian. It is the "largest" abelian group to which G maps, in the sense of a universal property.

It is usually denoted by G^ab or G_ab.

The abelianization of G coincides with the first homology group of G. In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper

Notes

In general the set of all commutators of the group is not a subgroup so we have to consider the subgroup generated by them. The smallest examples are two non-isomorphic groups of order 96. In each of these examples, the elements of the derived subgroup may be written as a product of two commutators.

The commutator subgroup can also be defined as the set of elements g of the group which have an expression as a product g=g₁g₂. . . g_k that can be rearranged to give the identity.

Examples

The commutator subgroup of the alternating group A₄ is the Klein four group. In Mathematics, an alternating group is the group of Even permutations of a Finite set. In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2
The commutator subgroup of the symmetric group $S n$ is the alternating group $A n$ . In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, an alternating group is the group of Even permutations of a Finite set.
The commutator subgroup of the quaternion group Q = {1, −1, i, −i, j, −j, k, −k} is [Q,Q]={1, −1}. In Group theory, the quaternion group is a non-abelian group of order 8

Properties

A group is abelian if and only if its commutator subgroup is the trivial group {e}. ↔ In Mathematics, a trivial group is a group consisting of a single element

Given a group G, a factor group G/N is abelian if and only if [G,G] ⊂ N.

If f : G → H is a group homomorphism, then $f ([G, G])$ is a subgroup of $[H, H]$ , because f maps commutators to commutators. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function This implies that the operation of forming derived groups is a functor from the category of groups to the category of groups. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

Applying this to endomorphisms of G, we find that $[G, G]$ is a fully characteristic subgroup of G, and in particular a normal subgroup of G. In Mathematics, a characteristic subgroup of a group G is a Subgroup H that is invariant under each Automorphism of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. (To reach the final conclusion, simply take conjugation with any particular g in G to be the automorphism in question. We see that g^-1 $[G, G]$ g = $[G, G]$ for every g in G, and therefore that $[G, G]$ is a normal subgroup of G. This is shown explicitly below). In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup

Category theory

In the language of category theory, abelian groups are a reflective subcategory of the category of groups, and abelianization is the reflector: the functor which assigns to every group its abelianization is left adjoint to the inclusion of abelian groups in groups. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a Subcategory A of a category B is said to be reflective in B when the Inclusion functor from In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

In other words, G^ab=G/[G,G] is the maximal abelian quotient of G. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G

In terms of universal properties, the commutator subgroup satisfies the following:

Given a group G, the commutator subgroup [G,G] is the uniquely defined subgroup^[2] of G so that given any homomorphism $f\colon G \to A$ from G to an abelian group A, then there exists a unique homomorphism $s\colon G/[G,G] \to A$ such that $s \circ \pi = f$ , where

π

is the abelianization $\pi\colon G \to G/[G,G]$ . In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism

Normality Property

The commutator subgroup is a normal subgroup. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. That is, $[G,G]\triangleleft G$ .

Proof: If g in G and h in [G,G] then $h g = h [h, g]$ is the product of two elements of [G,G].

Furthermore, if H is a subgroup of G that contains [G,G], then H is normal in G.

Proof: If g in G and h in H, then $h g = h [h, g]$ is the product of two elements of H.

Map from Out

Since the derived subgroup is characteristic, any automorphism of G induces an automorphism of the abelianization. In Mathematics, a characteristic subgroup of a group G is a Subgroup H that is invariant under each Automorphism of Since the abelianization is abelian, inner automorphisms act trivially, hence this yields a map

$\mbox{Out}(G) \to \mbox{Aut}(G^{\mbox{ab}})$

Footnotes

^ Some authors use the opposite convention of $[g, h]: = g h g - 1 h - 1$ ; this does not change the definition of commutator subgroup, as inverting switches between conventions: $[g - 1, h - 1] = g h g - 1 h - 1$ .
^ $[G, G]$ is a well-defined subgroup, since it is (fully) characteristic; it is not only "defined up to unique isomorphism", as is common in universal properties, as any isomorphism of G stabilizes it.

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