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(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