# Thompson transitivity theorem

Last updated

In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized by A. It originated in the proof of the odd order theorem by Feitand Thompson  ( 1963 ), where it was used to prove the Thompson uniqueness theorem.

In mathematics, especially group theory, the centralizer of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S are elements that satisfy a weaker condition. The centralizer and normalizer of S are subgroups of G, and can provide insight into the structure of G.

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. 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.

## Statement

Suppose that G is a finite group and p a prime such that all p-local subgroups are p-constrained. If A is a self-centralizing normal abelian subgroup of a p-Sylow subgroup such that A has rank at least 3, then the centralizer CG(A) act transitively on the maximal A-invariant q subgroups of G for any prime q  p.

## Related Research Articles In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below. These groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution. In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. In abstract algebra, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, an element of the set. In the case of a finite group, the set is finite. In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius.

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.

In mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then Op(G)Z(J ) is a normal subgroup of G, for any Sylow p-subgroupS.

In group theory, Bender's method is a method introduced by Bender (1970) for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify the Walter theorem on groups with abelian Sylow 2-subgroups Bender (1970b), and Gorenstein and Walter's classification of groups with dihedral Sylow 2-subgroups. Bender's method involves studying a maximal subgroup M containing the centralizer of an involution, and its generalized Fitting subgroup F*(M).

In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup. The idea is to try to construct a -subgroup of a finite group , which has a good chance of being normal in , by taking as generators certain -subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian -subgroups of The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including Gorenstein (1969) who defined signalizer functors, Glauberman (1976) who proved the Solvable Signalizer Functor Theorem for solvable groups, and McBride who proved it for all groups. This theorem is needed to prove the so-called "dichotomy" stating that a given nonabelian finite simple group either has local characteristic two, or is of component type. It thus plays a major role in the classification of finite simple groups.

In mathematical finite group theory, an N-group is a group all of whose local subgroups are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups.

In finite group theory, an area of abstract algebra, a strongly embedded subgroup of a finite group G is a proper subgroup H of even order such that H ∩ Hg has odd order whenever g is not in H. The Bender–Suzuki theorem, proved by Bender (1971) extending work of Suzuki (1962, 1964), classifies the groups G with a strongly embedded subgroup H. It states that either

1. G has cyclic or generalized quaternion Sylow 2-subgroups and H contains the centralizer of an involution
2. or G/O(G) has a normal subgroup of odd index isomorphic to one of the simple groups PSL2(q), Sz(q) or PSU3(q) where q≥4 is a power of 2 and H is O(G)NG(S) for some Sylow 2-subgroup S.

In mathematical group theory, a C-group is a group such that the centralizer of any involution has a normal Sylow 2-subgroup. They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups have trivial intersection.

In mathematical finite group theory, Thompson's original uniqueness theorem 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.

In mathematical finite group theory, the Dade isometry is an isometry from class functions on a subgroup H with support on a subset K of H to class functions on a group G. It was introduced by Dade (1964) as a generalization and simplification of an isometry used by Feit & Thompson (1963) in their proof of the odd order theorem, and was used by Peterfalvi (2000) in his revision of the character theory of the odd order theorem.

Everett Clarence Dade is a mathematician at University of Illinois at Urbana–Champaign working on finite groups and representation theory, who introduced the Dade isometry and Dade's conjecture.

In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step group in these two cases is slightly different.

In mathematical representation theory, coherence is a property of sets of characters that allows one to extend an isometry from the degree-zero subspace of a space of characters to the whole space. The general notion of coherence was developed by Feit, as a generalization of the proof by Frobenius of the existence of a Frobenius kernel of a Frobenius group and of the work of Brauer and Suzuki on exceptional characters. Feit & Thompson developed coherence further in the proof of the Feit–Thompson theorem that all groups of odd order are solvable.

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.

In mathematical group theory, the Thompson replacement theorem is a theorem about the existence of certain abelian subgroups of a p-group. The Glauberman replacement theorem is a generalization of it introduced by Glauberman.

George Glauberman is a mathematician at the University of Chicago who works on finite simple groups. He proved the ZJ theorem and the Z* theorem.

Cambridge University Press (CUP) is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world. It also holds letters patent as the Queen's Printer. The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.