Capable group

Definition
A group $$G$$ is said to be capable if it satisfies the following equivalent conditions:


 * 1) It is isomorphic to the defining ingredient::inner automorphism group of some group. In other words, there is a group $$H$$ such that $$G$$ is isomorphic to the quotient group $$H/Z(H)$$ where $$Z(H)$$ is the center of the group.
 * 2) Its defining ingredient::epicenter is the trivial group.

In terms of the image operator
The group property of being a capable group is obtained by applying the image operator for the quotient-defining function sending each group to its inner automorphism group.

Facts

 * The trivial group is capable; it occurs as the inner automorphism group of any abelian group
 * A nontrivial cyclic group cannot be capable. This is because there cannot be an element outside the center of a group, which, along with the center, generates the whole group.