N-group (finite group theory)
# 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.

