Exponent semigroup: Difference between revisions

Latest revision as of 19:04, 10 August 2012

Suppose $G$ is a group. The exponent semigroup of $G$ , denoted $E (G)$ is the following submonoid of the multiplicative monoid of integers:

$E (G) : = {n \in Z ∣ (x y)^{n} = x^{n} y^{n} \forall x, y \in G}$

In other words, it is the set of $n$ for which the $n^{t h}$ power map is an endomorphism (and hence a universal power endomorphism). For each such $n$ , we say that $G$ is a n-abelian group.

$E (G)$ is a multiplicative submonoid of $Z$ containing zero. In other words, it contains 0 and 1, and is closed under multiplication.
If $G$ has finite exponent $n$ , then $E (G)$ contains all multiples of $n$ . Moreover, $a \in E (G) ⟺ a + n \in E (G)$ (periodicity).
$E (G)$ is closed under reflection about $1 / 2$ , i.e., $a \in E (G) ⟺ 1 - a \in E (G)$ . This follows from n-abelian iff (1-n)-abelian.
$G$ is an abelian group if and only if its exponent semigroup is all of $Z$ . For one direction, see abelian implies universal power map is endomorphism. For the other direction, note that the element 2 is in $E (G)$ iff $G$ is abelian (square map is endomorphism iff abelian). Alternately, we can use that -1 is in $E (G)$ iff $G$ is abelian (inverse map is automorphism iff abelian).
Characterization of exponent semigroup of a finite p-group

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 * <math>\mathcal{E}(G)</math> is a multiplicative submonoid of <math>\mathbb{Z}</math> containing zero. In other words, it contains 0 and 1, and is closed under multiplication.
 * If <math>G</math> has finite [[exponent of a group|exponent]] <math>n</math>, then <math>\mathcal{E}(G)</math> contains all multiples of <math>n</math>. Moreover, <math>a \in \mathcal{E}(G) \iff a + n \in \mathcal{E}(G)</math> (periodicity).
+* <math>\mathcal{E}(G)</math> is closed under reflection about <math>1/2</math>, i.e., <math>a \in \mathcal{E}(G) \iff 1 - a \in \mathcal{E}(G)</math>. This follows from [[n-abelian iff (1-n)-abelian]].
 * <math>G</math> is an [[abelian group]] if and only if its exponent semigroup is all of <math>\mathbb{Z}</math>. For one direction, see [[abelian implies universal power map is endomorphism]]. For the other direction, note that the element 2 is in <math>\mathcal{E}(G)</math> iff <math>G</math> is abelian ([[square map is endomorphism iff abelian]]). Alternately, we can use that -1 is in <math>\mathcal{E}(G)</math> iff <math>G</math> is abelian ([[inverse map is automorphism iff abelian]]).
 * [[Characterization of exponent semigroup of a finite p-group]]