# 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 [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

[1] | J.G. Thompson, "Nonsolvable finite groups all of whose local subgroups are solvable" Bull. Amer. Math. Soc. , 74 (1968) pp. 383–437 |

[2a] | J.G. Thompson, "Nonsolvable finite groups all of whose local subgroups are solvable II" Pacific J. Math. , 33 (1970) pp. 451–536 |

[2b] | J.G. Thompson, "Nonsolvable finite groups all of whose local subgroups are solvable III" Pacific J. Math. , 39 (1971) pp. 483–534 |

[2c] | J.G. Thompson, "Nonsolvable finite groups all of whose local subgroups are solvable IV" Pacific J. Math. , 48 (1973) pp. 511–592 |

[2d] | J.G. Thompson, "Nonsolvable finite groups all of whose local subgroups are solvable V" Pacific J. Math. , 50 (1974) pp. 215–297 |

[2e] | J.G. Thompson, "Nonsolvable finite groups all of whose local subgroups are solvable VI" Pacific J. Math. , 51 (1974) pp. 573–630 |

**How to Cite This Entry:**

Minimal simple group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Minimal_simple_group&oldid=35920