N-group (finite group theory) explained

N-group (finite group theory) should not be confused with n-group (category theory).

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 groups. 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 groups PSL₂(q), PSL₃(3), the Suzuki groups Sz(2²ⁿ⁺¹), the unitary group U₃(3), the alternating group A₇, the Mathieu group M₁₁, and the Tits group. (The Tits group was overlooked in Thomson's original announcement in 1968, 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 groups 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 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 or the Tits group.

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.