Lesson plan
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) .
Article summary
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) .
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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