CN group

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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, Hall & Thompson 1961). 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.

[edit] References

• Burnside, William (1911), Theory of groups of finite order, pp. 503 (note M), ISBN 0486495752 (2004 reprinting) 
• 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: 1–17, MR0114856, ISSN 0025-5874, DOI 10.1007/BF01180468 
• Feit, Walter & Thompson, John G. (1963), “Solvability of groups of odd order”, Pacific Journal of Mathematics 13: 775–1029, MR0166261, 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 8: 686–695, MR0086818, ISSN 0002-9939, DOI 10.2307/2033280 
• Suzuki, Michio (1961), “Finite groups with nilpotent centralizers”, Transactions of the American Mathematical Society 99: 425–470, MR0131459, ISSN 0002-9947, DOI 10.2307/1993556