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 : 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. 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. The complete solution was given in, but further work on CN-groups was done in , 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(F2n) for n>1
 * The group PSL2(Fp) for p>3 a Fermat prime or Mersenne prime.
 * The group PSL2(F9)
 * The group PSL3(F4)