Exponent semigroup

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Definition

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

\mathcal{E}(G) := \{ n \in \mathbb{Z} \mid (xy)^n = x^ny^n \ \forall \ x,y \in G \}

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

Facts