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