# Difference between revisions of "Minimal simple group"

A non-Abelian simple group all proper subgroups of which are solvable (cf. Solvable group). A complete description of the finite minimal simple groups has been obtained (see , ), together with the classification of all finite groups whose local subgroups (that is, normalizers of $p$-subgroups) are solvable. Namely, a finite minimal simple group is isomorphic to one of the following projective special linear groups (cf. Projective group):

$\mathrm{PSL}(2,2^p)$, $p$ any prime;

$\mathrm{PSL}(2,3^m)$, $m$ any odd number;

$\mathrm{PSL}(2,p)$, $p \neq 3$ a prime satisfying $p \equiv 2,3 \pmod 5$;

$\mathrm{PSL}(3,3)$; or

the Suzuki group $\mathrm{Sz}(2^p)$, $p$ any odd prime. In particular, every finite minimal simple group is generated by two elements.

How to Cite This Entry:
Minimal simple group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_simple_group&oldid=13947
This article was adapted from an original article by S.P. Strunkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article