In group theory, a branch of mathematics, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power of p. In other words the group is a semidirect product of the normal p-complement and any Sylow p-subgroup. A group is called p-nilpotent if it has a normal p-complement.
Cayley showed that if the Sylow 2-subgroup of a group G is cyclic then the group has a normal 2-complement, which shows that the Sylow 2-subgroup of a simple group of even order cannot be cyclic.
Burnside ( 1911 , Theorem II, section 243) showed that if a Sylow p-subgroup of a group G is in the center of its normalizer then G has a normal p-complement. This implies that if p is the smallest prime dividing the order of a group G and the Sylow p-subgroup is cyclic, then G has a normal p-complement.
The Frobenius normal p-complement theorem is a strengthening of the Burnside normal p-complement theorem, which states that if the normalizer of every non-trivial subgroup of a Sylow p-subgroup of G has a normal p-complement, then so does G. More precisely, the following conditions are equivalent:
The Frobenius normal p-complement theorem shows that if every normalizer of a non-trivial subgroup of a Sylow p-subgroup has a normal p-complement then so does G. For applications it is often useful to have a stronger version where instead of using all non-trivial subgroups of a Sylow p-subgroup, one uses only the non-trivial characteristic subgroups. For odd primes p Thompson found such a strengthened criterion: in fact he did not need all characteristic subgroups, but only two special ones.
Thompson (1964) showed that if p is an odd prime and the groups N(J(P)) and C(Z(P)) both have normal p-complements for a Sylow P-subgroup of G, then G has a normal p-complement.
In particular if the normalizer of every nontrivial characteristic subgroup of P has a normal p-complement, then so does G. This consequence is sufficient for many applications.
The result fails for p = 2 as the simple group PSL2(F7) of order 168 is a counterexample.
Thompson (1960) gave a weaker version of this theorem.
Thompson's normal p-complement theorem used conditions on two particular characteristic subgroups of a Sylow p-subgroup. Glauberman improved this further by showing that one only needs to use one characteristic subgroup: the center of the Thompson subgroup.
Glauberman (1968) used his ZJ theorem to prove a normal p-complement theorem, that if p is an odd prime and the normalizer of Z(J(P)) has a normal p-complement, for P a Sylow p-subgroup of G, then so does G. Here Z stands for the center of a group and J for the Thompson subgroup.
The result fails for p = 2 as the simple group PSL2(F7) of order 168 is a counterexample.
In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.
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, 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, specifically group theory, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced by the group theorist Philip Hall.
In mathematics, a Ree group is a group of Lie type over a finite field constructed by Ree from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered.
In finite group theory, a branch of mathematics, the Thompson subgroup of a finite p-groupP refers to one of several characteristic subgroups of P. John G. Thompson originally defined to be the subgroup generated by the abelian subgroups of P of maximal rank. More often the Thompson subgroup is defined to be the subgroup generated by the abelian subgroups of P of maximal order or the subgroup generated by the elementary abelian subgroups of P of maximal rank. In general these three subgroups can be different, though they are all called the Thompson subgroup and denoted by .
In mathematics, George Glauberman's Z* theorem is stated as follows:
Z* theorem: Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G), which is the inverse image in G of the center of G/O(G).
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(S)) is a normal subgroup of G, for any Sylow p-subgroupS.
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups:
In mathematics, in the area of abstract algebra, a signalizer functor is a mapping from a potential finite subgroup to the centralizers of the nontrivial elements of an abelian group. The signalizer functor theorem provides the conditions under which the source of such a functor is in fact a subgroup.
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 mathematics, the Gorenstein–Walter theorem, proved by Gorenstein and Walter (1965a, 1965b, 1965c), states that if a finite group G has a dihedral Sylow 2-subgroup, and O(G) is the maximal normal subgroup of odd order, then G/O(G) is isomorphic to a 2-group, or the alternating group A7, or a subgroup of PΓL2(q) containing PSL2(q) for q an odd prime power. Note that A5 ≈ PSL2(4) ≈ PSL2(5) and A6 ≈ PSL2(9).
In mathematics, the Walter theorem, proved by John H. Walter, describes the finite groups whose Sylow 2-subgroup is abelian. Bender (1970) used Bender's method to give a simpler proof.
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.
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 Feit and Thompson, where it was used to prove the Thompson uniqueness theorem.
In finite group theory, a branch of mathematics, the Puig subgroup, introduced by Puig, 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.