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 -subgroups) are solvable. Namely, a finite minimal simple group is isomorphic to one of the following projective special linear groups:

, any prime;

, any odd number;

, a prime satisfying ;

; or

the Suzuki group , 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