Capable group

In mathematics, in the realm of group theory, a group is said to be capable if it is isomorphic to the quotient G / Z(G) of some group G 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, ..., n_k where n_i divides n_i +1 and n_k −1 = n_k.

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 $G\to \mathrm{Inn}(G)$ has kernel Z(G); by the first isomorphism theorem, G / Z(G) is equivalent to $\mathrm{Inn}(G)$.

References

• Baer, Reinhold (1938), "Groups with preassigned central and central quotient group", Transactions of the American Mathematical Society 44 (3): 387–412, doi:10.2307/1989887 

External links

• Bounds on the index of the center in capable groups