Difference between revisions of "Minimal simple group"
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− | A non-Abelian [[ | + | {{TEX|done}} |
+ | 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 [[ | + | the [[Suzuki group]] $\mathrm{Sz}(2^p)$, $p$ any odd prime. In particular, every finite minimal simple group is generated by two elements. |
====References==== | ====References==== | ||
− | <table><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></table> | + | <table> |
+ | <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> | ||
+ | </table> |
Latest revision as of 19:25, 28 December 2014
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 |
Minimal simple group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_simple_group&oldid=35920