# CN-group

In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of (Burnside 1911): are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are solvable (Suzuki 1957). Further progress was made showing that **CN-groups**, groups in which the centralizer of a non-identity element is nilpotent, of odd order are solvable (Feit Thompson). The complete solution was given in (Feit Thompson), but further work on CN-groups was done in (Suzuki 1961), giving more detailed information about the structure of these groups. For instance, a non-solvable CN-group *G* is such that its largest solvable normal subgroup *O*_{∞}(*G*) is a 2-group, and the quotient is a group of even order.

## Examples

Solvable CN groups include

- Nilpotent groups
- Frobenius groups whose Frobenius complement is nilpotent
- 3-step groups, such as the symmetric group
*S*_{4}

Non-solvable CN groups include:

- The Suzuki simple groups
- The groups PSL
_{2}(**F**_{2n}) for*n*>1 - The group PSL
_{2}(**F**_{p}) for*p*>3 a Fermat prime or Mersenne prime. - The group PSL
_{2}(**F**_{9}) - The group PSL
_{3}(**F**_{4})

## References

- Burnside, William (2004),
*Theory of groups of finite order*, Dover Publications, pp. 503 (note M), ISBN 978-0-486-49575-0 - Feit, Walter; Thompson, John G.; Hall, Marshall Jr. (1960), "Finite groups in which the centralizer of any non-identity element is nilpotent",
*Math. Z.***74**(1): 1–17, doi:10.1007/BF01180468 - Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order",
*Pacific Journal of Mathematics***13**: 775–1029, doi:10.2140/pjm.1963.13.775, ISSN 0030-8730, http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.pjm&issue=1103053941 - Suzuki, Michio (1957), "The nonexistence of a certain type of simple groups of odd order",
*Proceedings of the American Mathematical Society*(American Mathematical Society)**8**(4): 686–695, doi:10.2307/2033280 - Suzuki, Michio (1961), "Finite groups with nilpotent centralizers",
*Transactions of the American Mathematical Society*(American Mathematical Society)**99**(3): 425–470, doi:10.2307/1993556

Original source: https://en.wikipedia.org/wiki/CN-group.
Read more |