CN-group

This article is about mathematics. For the sports car classification, see Group CN. For the media group, see CN Group. For the Canadian corporate group, see Canadian National. For the cyano-group, see cyanide.

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 & Hall 1960). The complete solution was given in (Feit & Thompson 1963), 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) [1911], 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, MR 0114856, S2CID 120550114
• 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, MR 0166261
• Suzuki, Michio (1957), "The nonexistence of a certain type of simple groups of odd order", Proceedings of the American Mathematical Society, 8 (4), American Mathematical Society: 686–695, doi:10.2307/2033280, JSTOR 2033280, MR 0086818
• Suzuki, Michio (1961), "Finite groups with nilpotent centralizers", Transactions of the American Mathematical Society, 99 (3), American Mathematical Society: 425–470, doi:10.2307/1993556, JSTOR 1993556, MR 0131459