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A non-Abelian [[Simple group|simple group]] all proper subgroups of which are solvable (cf. [[Solvable group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m0639001.png" />-subgroups) are solvable. Namely, a finite minimal simple group is isomorphic to one of the following projective special linear groups:
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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]]):
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m0639002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m0639003.png" /> any prime;
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$\mathrm{PSL}(2,2^p)$, $p$ any prime;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m0639004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m0639005.png" /> any odd number;
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$\mathrm{PSL}(2,3^m)$, $m$ any odd number;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m0639006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m0639007.png" /> a prime satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m0639008.png" />;
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$\mathrm{PSL}(2,p)$, $p \neq 3$ a prime satisfying $p \equiv 2,3 \pmod 5$;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m0639009.png" />; or
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$\mathrm{PSL}(3,3)$; or
  
the [[Suzuki group|Suzuki group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m06390010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063900/m06390011.png" /> any odd prime. In particular, every finite minimal simple group is generated by two elements.
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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>
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<table>
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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>
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<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>
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<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>
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<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>
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<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>
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<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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</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
How to Cite This Entry:
Minimal simple group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_simple_group&oldid=35920
This article was adapted from an original article by S.P. Strunkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article