Difference between revisions of "Talk:Sporadic simple group"
From Encyclopedia of Mathematics
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|- | |- | ||
| $M_{11}$ | | $M_{11}$ | ||
| − | | | + | | rowspan="5" | Mathieu groups |
| $2^4.3^2.5.11$ | | $2^4.3^2.5.11$ | ||
|- | |- | ||
| $M_{12}$ | | $M_{12}$ | ||
| − | |||
| $2^6.3^3.5.11$ | | $2^6.3^3.5.11$ | ||
|- | |- | ||
| $M_{22}$ | | $M_{22}$ | ||
| − | |||
| $2^7.3^2.5.7.11$ | | $2^7.3^2.5.7.11$ | ||
|- | |- | ||
| $M_{23}$ | | $M_{23}$ | ||
| − | |||
| $2^7.3^2.5.7.11.23$ | | $2^7.3^2.5.7.11.23$ | ||
|- | |- | ||
| $M_{24}$ | | $M_{24}$ | ||
| − | |||
| $2^{10}.3^3.5.7.11.23$ | | $2^{10}.3^3.5.7.11.23$ | ||
|- | |- | ||
| Line 43: | Line 39: | ||
|- | |- | ||
| $Co_1$ | | $Co_1$ | ||
| − | | | + | | rowspan="3" | Conway groups |
| $2^{21}.3^9.5^4.7^2.11.13.23$ | | $2^{21}.3^9.5^4.7^2.11.13.23$ | ||
|- | |- | ||
| $Co_2$ | | $Co_2$ | ||
| − | |||
| $2^{18}.3^6.5^3.7.11.23$ | | $2^{18}.3^6.5^3.7.11.23$ | ||
|- | |- | ||
| $Co_3$ | | $Co_3$ | ||
| − | |||
| $2^{10}.3^7.5^3.7.11.23$ | | $2^{10}.3^7.5^3.7.11.23$ | ||
|- | |- | ||
| $F_{22}$, $M(22)$ | | $F_{22}$, $M(22)$ | ||
| − | | | + | | rowspan="3" | Fischer groups |
| $2^{17}.3^9.5^2.7.11.13$ | | $2^{17}.3^9.5^2.7.11.13$ | ||
|- | |- | ||
| $F_{23}$, $M(23)$ | | $F_{23}$, $M(23)$ | ||
| − | |||
| $2^{18}.3^{13}.5^2.7.11.13.17.23$ | | $2^{18}.3^{13}.5^2.7.11.13.17.23$ | ||
|- | |- | ||
| $F_{24}^\prime$, $M(24)^\prime$ | | $F_{24}^\prime$, $M(24)^\prime$ | ||
| − | |||
| $2^{21}.3^{16}.5^2.7^3.11.13.17.23.29$ | | $2^{21}.3^{16}.5^2.7^3.11.13.17.23.29$ | ||
|- | |- | ||
| Line 71: | Line 63: | ||
|- | |- | ||
| $He$, $HHM$ | | $He$, $HHM$ | ||
| − | | Held–Higman–McKay | + | | Held–Higman–McKay group |
| $2^{10}.2^3.5^2.7^3.17$ | | $2^{10}.2^3.5^2.7^3.17$ | ||
|- | |- | ||
| Line 91: | Line 83: | ||
|- | |- | ||
| $O'N$, $O'NS$ | | $O'N$, $O'NS$ | ||
| − | | O'Nan–Sims | + | | O'Nan–Sims group |
| $2^9.3^4.5.7^3.11.19.31$ | | $2^9.3^4.5.7^3.11.19.31$ | ||
|- | |- | ||
| Line 113: | Line 105: | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.A. Syskin, "Abstract properties of the simple sporadic groups" ''Russian Math. Surveys'' , '''35''' : 5 (1980) pp. 209–246 ''Uspekhi Mat. Nauk'' , '''35''' : 5 (1980) pp. 181–212</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Aschbacher, "The finite simple groups and their classification" , Yale Univ. Press (1980)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.A. Syskin, "Abstract properties of the simple sporadic groups" ''Russian Math. Surveys'' , '''35''' : 5 (1980) pp. 209–246 ''Uspekhi Mat. Nauk'' , '''35''' : 5 (1980) pp. 181–212</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Aschbacher, "The finite simple groups and their classification" , Yale Univ. Press (1980)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | The recent classification of the finite simple groups (1981) has led to the conclusion that — up to a uniqueness proof for the Monster as the only simple group of its order with certain additional properties — every non-Abelian finite simple group is isomorphic to: an [[Alternating group|alternating group]] on at least | + | The recent classification of the finite simple groups (1981) has led to the conclusion that — up to a uniqueness proof for the Monster as the only simple group of its order with certain additional properties — every non-Abelian finite simple group is isomorphic to: an [[Alternating group|alternating group]] on at least 5 letters, a group of (twisted or untwisted) Lie type, or one of the above 26 sporadic groups. See [[#References|[a2]]] for a discussion of the proof. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups" , Clarendon Press (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D. Gorenstein, "Finite simple groups. An introduction to their classification" , Plenum (1982)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups" , Clarendon Press (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D. Gorenstein, "Finite simple groups. An introduction to their classification" , Plenum (1982)</TD></TR></table> | ||
Revision as of 16:42, 30 April 2012
A simple finite group that does not belong to any of the known infinite series of simple finite groups. The twenty-six sporadic simple groups are listed in the following table.
| notation | name | order |
|---|---|---|
| $M_{11}$ | Mathieu groups | $2^4.3^2.5.11$ |
| $M_{12}$ | $2^6.3^3.5.11$ | |
| $M_{22}$ | $2^7.3^2.5.7.11$ | |
| $M_{23}$ | $2^7.3^2.5.7.11.23$ | |
| $M_{24}$ | $2^{10}.3^3.5.7.11.23$ | |
| $J_1$ | Janko group | $2^3.3.5.7.11.19$ |
| $J_2$, $HJ$ | Hall–Janko group | $2^7.3^3.5^2.7$ |
| $J_3$, $HJM$ | Hall–Janko–McKay group | $2^7.3^5.5.17.19$ |
| $J_4$ | Janko group | $2^{21}.3^3.5.7.11^3.23.29.31.37.43$ |
| $Co_1$ | Conway groups | $2^{21}.3^9.5^4.7^2.11.13.23$ |
| $Co_2$ | $2^{18}.3^6.5^3.7.11.23$ | |
| $Co_3$ | $2^{10}.3^7.5^3.7.11.23$ | |
| $F_{22}$, $M(22)$ | Fischer groups | $2^{17}.3^9.5^2.7.11.13$ |
| $F_{23}$, $M(23)$ | $2^{18}.3^{13}.5^2.7.11.13.17.23$ | |
| $F_{24}^\prime$, $M(24)^\prime$ | $2^{21}.3^{16}.5^2.7^3.11.13.17.23.29$ | |
| $HS$ | Higman–Sims group | $2^9.3^2.5^3.7.11$ |
| $He$, $HHM$ | Held–Higman–McKay group | $2^{10}.2^3.5^2.7^3.17$ |
| $Suz$ | Suzuki group | $2^{13}.3^7.5^2.7.11.13$ |
| $M^c$ | McLaughlin group | $2^7.3^6.5^3.7.11$ |
| $Ly$ | Lyons group | $2^8.3^7.5^6.7.11.31.37.67$ |
| $Ru$ | Rudvalis group | $2^{14}.3^3.5^3.7.13.29$ |
| $O'N$, $O'NS$ | O'Nan–Sims group | $2^9.3^4.5.7^3.11.19.31$ |
| $F_1$, $M$ | Monster, Fischer–Griess group | $2^{46}.3^{20}.5^9.7^6.11^2.13^3.17.19.23.29.31.41.47.59.71$ |
| $F_2$, $B$ | Baby monster | $2^{41}.3^{13}.5^6.7^2.11.13.17.19.23.31.47$ |
| $F_3$, $E$, $Th$ | Thompson group | $2^{15}.3^{10}.5^3.7^2.13.19.31$ |
| $F_5$, $D$, $HN$ | Harada–Norton group | $2^{14}.3^6.5^6.7.11.19$ |
References
| [1] | S.A. Syskin, "Abstract properties of the simple sporadic groups" Russian Math. Surveys , 35 : 5 (1980) pp. 209–246 Uspekhi Mat. Nauk , 35 : 5 (1980) pp. 181–212 |
| [2] | M. Aschbacher, "The finite simple groups and their classification" , Yale Univ. Press (1980) |
Comments
The recent classification of the finite simple groups (1981) has led to the conclusion that — up to a uniqueness proof for the Monster as the only simple group of its order with certain additional properties — every non-Abelian finite simple group is isomorphic to: an alternating group on at least 5 letters, a group of (twisted or untwisted) Lie type, or one of the above 26 sporadic groups. See [a2] for a discussion of the proof.
References
| [a1] | J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups" , Clarendon Press (1985) |
| [a2] | D. Gorenstein, "Finite simple groups. An introduction to their classification" , Plenum (1982) |
How to Cite This Entry:
Sporadic simple group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sporadic_simple_group&oldid=25778
Sporadic simple group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sporadic_simple_group&oldid=25778