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 S₄

Non-solvable CN groups include:

• The Suzuki simple groups
• The groups PSL₂(F_2ⁿ) for n>1
• The group PSL₂(F_p) for p>3 a Fermat prime or Mersenne prime.
• The group PSL₂(F₉)
• The group PSL₃(F₄)

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