# N-group (finite group 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 Thompson (1968, 1970, 1971, 1973, 1974, 1974b) in a series of 6 papers totaling about 400 pages.
The simple N-groups consist of the special linear groups PSL$_2$(q), PSL$_3$(3), the Suzuki groups Sz(2$^{2n+1}$), the unitary group U$_3$(3), the alternating group A$_7$, the Mathieu group M$_{11}$, 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.
Gorenstein & Lyons (1976) 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 U$_3$(q).

## Proof

Gorenstein (1980, 16.5) gives a summary of Thompson's classification of N-groups.
The primes dividing the order of the group are divided into four classes π$_1$, π$_2$, π$_3$, π$_4$ as follows
* π$_1$ is the set of primes p such that a Sylow p-subgroup is nontrivial and cyclic.
* π$_2$ is the set of primes p such that a Sylow p-subgroup P is non-cyclic but SCN$_3$(P) is empty
* π$_3$ is the set of primes p such that a Sylow p-subgroup P has SCN$_3$(P) nonempty and normalizes a nontrivial abelian subgroup of order prime to p.
* π$_4$ is the set of primes p such that a Sylow p-subgroup P has SCN$_3$(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.
* Thompson (1968) Gives a general introduction, stating the main theorem and proving many preliminary lemmas.
* Thompson (1970) characterizes the groups E$_2$(3) and S$_4$(3) (in Thompson's notation; these are the exceptional group G$_2$(3) and the symplectic group Sp$_4$(3)) which are not N-groups but whose characterizations are needed in the proof of the main theorem.
* Thompson (1971) covers the case where 2∉π$_4$. Theorem 11.2 shows that if 2∈π$_2$ then the group is PSL$_2$(q), M$_{11}$, A$_7$, U$_3$(3), or PSL$_3$(3). The possibility that 2∈π$_3$ 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.
* Thompson (1973) and Thompson (1974) cover the cases when 2∈π$_4$ 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 E$_2$(3) and S$_4$(3) in his second paper, which are not N-groups.
* Thompson (1974) covers the case when 2∈π$_4$ 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 Thompson (1968, corollary 1)
* PSL$_2$(2$^p$), p a prime.
* PSL$_2$(3$^p$), p an odd prime.
* PSL$_2$(p), p > 3 a prime congruent to 2 or 3 mod 5
* Sz(2$^p$), p an odd prime.
* PSL$_3$(3)
In other words a non-cyclic finite simple group must have a subquotient isomorphic to one of these groups.