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.
The simple N-groups were classified by John 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 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.
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 U3(q).
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 π1, π2, π3, π4 as follows
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.
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)
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