View source for Minimal simple group

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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 [[#References|[1]]], ), 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.

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.G. Thompson,   "Nonsolvable finite groups all of whose local subgroups are solvable"  ''Bull. Amer. Math. Soc.'' , '''74'''  (1968)  pp. 383–437</TD></TR>
<TR><TD valign="top">[2a]</TD> <TD valign="top">  J.G. Thompson,   "Nonsolvable finite groups all of whose local subgroups are solvable II"  ''Pacific J. Math.'' , '''33'''  (1970)  pp. 451–536</TD></TR>
<TR><TD valign="top">[2b]</TD> <TD valign="top">  J.G. Thompson,   "Nonsolvable finite groups all of whose local subgroups are solvable III"  ''Pacific J. Math.'' , '''39'''  (1971)  pp. 483–534</TD></TR>
<TR><TD valign="top">[2c]</TD> <TD valign="top">  J.G. Thompson,   "Nonsolvable finite groups all of whose local subgroups are solvable IV"  ''Pacific J. Math.'' , '''48'''  (1973)  pp. 511–592</TD></TR>
<TR><TD valign="top">[2d]</TD> <TD valign="top">  J.G. Thompson,   "Nonsolvable finite groups all of whose local subgroups are solvable V"  ''Pacific J. Math.'' , '''50'''  (1974)  pp. 215–297</TD></TR>
<TR><TD valign="top">[2e]</TD> <TD valign="top">  J.G. Thompson,   "Nonsolvable finite groups all of whose local subgroups are solvable VI"  ''Pacific J. Math.'' , '''51'''  (1974)  pp. 573–630</TD></TR>
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