Abelization

The Abelisierung (also Abelianisierung or Faktorkommutatorgruppe ) is a construction of the mathematical part area of group theory . Abelizing a group is in some ways the best approximation by an Abelian group .

definition

The factor group

${\displaystyle G^{\mathrm {ab} }=G/K(G)}$

a group after its commutator subgroup is called the abelization of . The term abelization is also used for canonical surjection ${\displaystyle G}$ ${\displaystyle K(G)}$ ${\displaystyle G}$

${\displaystyle G\to G^{\mathrm {ab} }}$

used.

properties

• Abelization is an Abelian group; the abelization of an Abelian group is the group itself.
• If a group homomorphism, the concatenation induces a canonical homomorphism ; the abelization is functional. ${\displaystyle G_{1}\to G_{2}}$ ${\displaystyle G_{1}\to G_{2}\to G_{2}^{\mathrm {ab} }}$ ${\displaystyle G_{1}^{\mathrm {ab} }\to G_{2}^{\mathrm {ab} }}$
• The Abelization is left adjoint to the forget function from the category of the Abelian groups to the category of all groups, i.e. H. is any group and an Abelian group, the canonical mapping induces a bijection ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle G\to G^{\mathrm {ab} }}$

${\displaystyle \operatorname {Hom} (G^{\mathrm {ab} },A)\cong \operatorname {Hom} (G,A).}$

In other words: Every homomorphism into an Abelian group factors via the Abelization.

• Have particular and the same characters . ${\displaystyle G}$ ${\displaystyle G^{\mathrm {ab} }}$
• The abelization of a group is canonically dual to group cohomology ${\displaystyle G}$

${\displaystyle H^{2}(G,\mathbb {Z} )\cong H^{1}(G,\mathbb {Q} /\mathbb {Z} )\cong \operatorname {Hom} (G,\mathbb {Q} /\mathbb {Z} ).}$

Examples

• If a simple group is not Abelian, then its Abelization is the trivial group .
• For a well - dotted path-connected topological space , the first homology group is the abelization of the fundamental group . ${\displaystyle X}$ ${\displaystyle H_{1}(X,\mathbb {Z} )}$
• The class field theory deals with the description of the Abelization of the absolute Galois group of a number field . ${\displaystyle \mathrm {Gal} ({\bar {K}}/K)}$ ${\displaystyle K}$

Relocation

If a subgroup is a finite group , there is a canonical homomorphism ${\displaystyle H}$ ${\displaystyle G}$

${\displaystyle \operatorname {Ver} \colon G^{\mathrm {ab} }\to H^{\mathrm {ab} },}$

called the relocation . It is dual to the correction

${\displaystyle \operatorname {cor} \colon H^{2}(H,\mathbb {Z} )\to H^{2}(G,\mathbb {Z} ),}$

but can also be described explicitly: Let it be a section of the canonical projection (no homomorphism, just a mapping). Then the shift is given by ${\displaystyle s\colon H\backslash G\to G}$

${\displaystyle \operatorname {Ver} (gK(G))=\prod _{c\in H\backslash G}s(c)gs(cg)^{-1}K(H)\qquad (g\in G).}$

swell

1. ↑ JP May: A Concise Course in Algebraic Topology . University of Chicago Press, Chicago 1999. ISBN 0-226-51183-9 : Sections 14.4 and 15.1
2. ^ J. Neukirch, A. Schmidt, K. Wingberg: Cohomology of number fields . Springer-Verlag, Berlin-Heidelberg-New York 1999, ISBN 3-540-66671-0 : Section I.5, p. 52f.