N-group Theorem

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(2^2''n''+1), the unitary group U₃(3), the alternating group ''A''₇, 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 ''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.
* covers the case where 2∉π₄. Theorem 11.2 shows that if 2∈π₂ then the group is PSL₂(''q''), M₁₁, ''A''₇, 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 ''E''₂(3) and ''S''₄(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.

References

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*{{Citation , last1=Thompson , first1=John G. , author1-link=John G. Thompson , title=Nonsolvable finite groups all of whose local subgroups are solvable. VI , url=http://projecteuclid.org/euclid.pjm/1102912481 , mr=0369512 , year=1974b , journal=Pacific Journal of Mathematics , issn=0030-8730 , volume=51 , issue=2 , pages=573–630 , doi=10.2140/pjm.1974.51.573, doi-access=free

Finite groups