N-abelian group: Difference between revisions

Revision as of 23:03, 7 August 2012

Definition

Suppose $n$ is an integer. A group $G$ is termed a $n$ -abelian group if the $n^{t h}$ power map $x \mapsto x^{n}$ is an endomorphism of $G$ , i.e., $(x y)^{n} = x^{n} y^{n}$ for all $x, y \in G$ . If this is the case, then the $n^{t h}$ power map is termed a universal power endomorphism of $G$ .

The set of $n$ for which $G$ is $n$ -abelian is termed the exponent semigroup of $G$ . It is a submonoid of the multiplicative monoid of integers.

Facts

Every group is 0-abelian and 1-abelian.
Abelian implies n-abelian for all n
2-abelian iff abelian
-1-abelian iff abelian
n-abelian implies every nth power and (n-1)th power commute
n-abelian implies n(n-1)-central

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 Suppose <math>n</math> is an integer. A [[group]] <math>G</math> is termed a '''<math>n</math>-abelian group''' if the <math>n^{th}</math> power map <math>x \mapsto x^n</math> is an [[endomorphism]] of <math>G</math>, i.e., <math>(xy)^n = x^ny^n</math> for all <math>x,y \in G</math>. If this is the case, then the <math>n^{th}</math> power map is termed a [[universal power endomorphism]] of <math>G</math>.
+The set of <math>n</math> for which <math>G</math> is <math>n</math>-abelian is termed the [[exponent semigroup]] of <math>G</math>. It is a submonoid of the multiplicative monoid of integers.
+==Facts==
+* Every group is 0-abelian and 1-abelian.
+* [[Abelian implies n-abelian for all n]]
+* [[2-abelian iff abelian]]
+* [[-1-abelian iff abelian]]
+* [[n-abelian implies every nth power and (n-1)th power commute]]
+* [[n-abelian implies n(n-1)-central]]