In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable group

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...

s. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups.

Simple N-groups

The simple N-groups were classified by in a series of 6 papers totaling about 400 pages.

The simple N-groups consist of the special linear group

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

s PSL₂(q),PSL₃(3), the Suzuki groups Sz(2²ⁿ⁺¹), the unitary group U₃(3), the alternating group A₇, the Mathieu group

Mathieu group

In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...

 M₁₁, and the Tits group

Tits group

In mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by ....

. (The Tits group was overlooked in Thomson's original announcement in 1963, which was made before the discovery of the Tits group, but Hearn pointed out that it was also a simple N-group.) More generally Thompson showed that any non-solvable N-group is a subgroup of Aut(G) containing G for some simple N-group G.

generalized Thompson's theorem to the case of groups where all 2-local subgroups are solvable. The only extra simple groups that appear are the unitary group U₃(q).

Proof

gives a summary of Thompson's classification of N-groups.

The primes dividing the order of the group are divided into four classes π₁, π₂, π₃, π₄ as follows

• π₁ is the set of primes p such that a Sylow p-subgroup is nontrivial and cyclic.
• π₂ is the set of primes p such that a Sylow p-subgroup P is non-cyclic but SCN₃(P) is empty
• π₃ is the set of primes p such that a Sylow p-subgroup P has SCN₃(P) nonempty and normalizes a nontrivial abelian subgroup of order prime to p.
• π₄ is the set of primes p such that a Sylow p-subgroup P has SCN₃(P) nonempty but does not normalize a nontrivial abelian subgroup of order prime to p.

The proof is subdivided into several cases depending on which of these four classes the prime 2 belongs to, and also on an integer e, which is the largest integer for which there is an elementary abelian subgroup of rank e normalized by a nontrivial 2-subgroup intersecting it trivially.
Gives a general introduction, stating the main theorem and proving many preliminary lemmas. characterizes the groups E₂(3) and S₄(3) (in Thompson's notation; these are the exceptional group G₂(3) and the symplectic group Sp₄(3)) which are not N-groups but whose characterizations are needed in the proof of the main theorem. covers the case where 2∉π₄. Theorem 11.2 shows that if 2∈π₂ then the group is PSL₂(q), M₁₁, A₇, U₃(3), or PSL₃(3). The possibility that 2∈π₃ is ruled out by showing that any such group must be a C-group and using Suzuki's classification of C-groups to check that none of the groups found by Suzuki satisfy this condition. and cover the cases when 2∈π₄ and e≥3, or e=2. He shows that either G is a C-group

C-group

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.The simple...

 so a Suzuki group, or satisfies his characterization of the groups E₂(3) and S₄(3) in his second paper, which are not N-groups. covers the case when 2∈π₄ and e=1, where the only possibilities are that G is a C-group

C-group

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.The simple...

 or the Tits group

Tits group

In mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by ....

Consequences

A minimal simple group is a non-cyclic simple group all of whose proper subgroups are solvable.
The complete list of minimal finite simple groups is given as follows

• PSL₂(2^p), p a prime.
• PSL₂(3^p), p an odd prime.
• PSL₂(p), p > 3 a prime congruent to 2 or 3 mod 5
• Sz(2^p), p an odd prime.
• PSL₃(3)

In other words a non-cyclic finite simple group must have a subquotient isomorphic to one of these groups.