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 n 1 , ..., 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 \operatorname{Inn}(G) has kernel ; by the first isomorphism theorem, is equivalent to \operatorname{Inn}(G) .

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 1 , ..., 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 \operatorname{Inn}(G) has kernel ; by the first isomorphism theorem, is equivalent to \operatorname{Inn}(G) .

External links

Bounds on the index of the center in capable groups

Text derived from https://en.wikipedia.org/w/index.php?oldid=1329017951 under CC BY-SA 4.0; modifications have been made.