In mathematics, in the realm of group theory, a group is said to be capable if it is isomorphic to the quotient of some group by its center.
These groups were first studied by Reinhold Baer, who showed that a finite abelian group is capable if and only if it is a product of cyclic groups of orders n<sub>1</sub>, ..., n<sub>k</sub> where n<sub>i</sub> divides n<sub>i+1</sub> and n<sub>kâÂÂ1</sub> = n<sub>k</sub>.
An equivalent condition for a group to be capable is if it occurs as the inner automorphism group of some group. To see this, note that the canonical surjective map has kernel ; by the first isomorphism theorem, is equivalent to .