Capable group
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 n1,..., nk where ni divides ni+1 and nk−1 = nk.
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.