Vipul: New page: {{definition equivalence|central subgroup}} ==Statement== The following are equivalent for a subgroup $H$ of a group $G$ : # $H$ is a [[central subgroup]...

2008-09-16T21:10:31Z

New page: {{definition equivalence|central subgroup}} ==Statement== The following are equivalent for a subgroup <math>H</math> of a group <math>G</math>: # <math>H</math> is a [[central subgroup]...

New page

{{definition equivalence|central subgroup}}

==Statement==

The following are equivalent for a subgroup <math>H</math> of a group <math>G</math>:

# <math>H</math> is a [[central subgroup]] of <math>G</math>: in other words, <math>H \le Z(G)</math>, or every element of <math>H</math> commutes with every element of <math>G</math>.
# <math>H</math> is an [[Abelian group]], and is a [[central factor]] of <math>G</math>.

==Definitions used==

===Central factor===

A subgroup <math>H</math> of a group <math>G</math> is termed a '''central factor''' if it satisfies the following equivalent conditions:

# <math>HC_G(H) = G</math>.
# Every inner automorphism of <math>G</math> restricts to an inner automorphism of <math>H</math>. The function restriction expression for this is:

Inner automorphism <math>\to</math> Inner automorphism

===Central subgroup===

A subgroup <math>H</math> of a group <math>G</math> is termed a '''central subgroup''' if it satisfies the following equivalent conditions:

# <math>C_G(H) = G</math>.
# Every inner automorphism of <math>G</math> restricts to the identity map on <math>H</math>. The function restriction expression for this is:

Inner automorphism <math>\to</math> Identity map

==Proof==

===Proof in the language of centralizers===

{{fillin}}

{{frexp implication}}

Abelian central factor equals central subgroup - Revision history

Vipul: New page: {{definition equivalence|central subgroup}} ==Statement== The following are equivalent for a subgroup H of a group G: # H is a [[central subgroup]...

Vipul: New page: {{definition equivalence|central subgroup}} ==Statement== The following are equivalent for a subgroup $H$ of a group $G$ : # $H$ is a [[central subgroup]...