Thompson uniqueness theorem

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In mathematical finite group theory, Thompson's original uniqueness theorem( Feit & Thompson 1963 , theorems 24.5 and 25.2) states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank  3. Bender (1973) gave a shorter proof of the uniqueness theorem.

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John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize.

In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson.

In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theorem and the classification of finite simple groups. Several important infinite groups are CA-groups, such as free groups, Tarski monsters, and some Burnside groups, and the locally finite CA-groups have been classified explicitly. CA-groups are also called commutative-transitive groups because commutativity is a transitive relation amongst the non-identity elements of a group if and only if the group is a CA-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.

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In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by Walter Feit and John G. Thompson (1962). The conjecture states that there are no distinct prime numbers p and q such that

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In finite group theory, a p-stable group for an odd prime p is a finite group satisfying a technical condition introduced by Gorenstein and Walter in order to extend Thompson's uniqueness results in the odd order theorem to groups with dihedral Sylow 2-subgroups.

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In mathematical finite group theory, the Puig subgroup, introduced by Puig (1976), is a characteristic subgroup of a p-group analogous to the Thompson subgroup.