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 & 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

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