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)
References