Difference between revisions of "Sporadic simple group"
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| − | A [[Simple finite group|simple finite group]] that does not belong to any of the known infinite series of simple finite groups. The | + | A ''sporadic simple group'' is |
| + | a [[Simple finite group|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. | ||
| + | $\def\d{\cdot}$ | ||
{| class="wikitable" style="margin: 1em auto 1em auto;" | {| class="wikitable" style="margin: 1em auto 1em auto;" | ||
|+ The twenty-six sporadic simple groups | |+ The twenty-six sporadic simple groups | ||
| − | ! | + | ! Notation |
| − | ! | + | ! Name |
| − | ! | + | ! Order |
|- | |- | ||
| $M_{11}$ | | $M_{11}$ | ||
| − | | rowspan="5" | Mathieu groups | + | | rowspan="5" | [[Mathieu group | Mathieu groups]] |
| − | | $2^4 | + | | $2^4\d 3^2\d 5\d 11$ |
|- | |- | ||
| $M_{12}$ | | $M_{12}$ | ||
| − | | $2^6 | + | | $2^6\d 3^3\d 5\d 11$ |
|- | |- | ||
| $M_{22}$ | | $M_{22}$ | ||
| − | | $2^7 | + | | $2^7\d 3^2\d 5\d 7\d 11$ |
|- | |- | ||
| $M_{23}$ | | $M_{23}$ | ||
| − | | $2^7 | + | | $2^7\d 3^2\d 5\d 7\d 11\d 23$ |
|- | |- | ||
| $M_{24}$ | | $M_{24}$ | ||
| − | | $2^{10} | + | | $2^{10}\d 3^3\d 5\d 7\d 11\d 23$ |
|- | |- | ||
| $J_1$ | | $J_1$ | ||
| Janko group | | Janko group | ||
| − | | $2^3 | + | | $2^3\d 3\d 5\d 7\d 11\d 19$ |
|- | |- | ||
| $J_2$, $HJ$ | | $J_2$, $HJ$ | ||
| Hall–Janko group | | Hall–Janko group | ||
| − | | $2^7 | + | | $2^7\d 3^3\d 5^2\d 7$ |
|- | |- | ||
| $J_3$, $HJM$ | | $J_3$, $HJM$ | ||
| Hall–Janko–McKay group | | Hall–Janko–McKay group | ||
| − | | $2^7 | + | | $2^7\d 3^5\d 5\d 17\d 19$ |
|- | |- | ||
| $J_4$ | | $J_4$ | ||
| Janko group | | Janko group | ||
| − | | $2^{21} | + | | $2^{21}\d 3^3\d 5\d 7\d 11^3\d 23\d 29\d 31\d 37\d 43$ |
|- | |- | ||
| $Co_1$ | | $Co_1$ | ||
| rowspan="3" | Conway groups | | rowspan="3" | Conway groups | ||
| − | | $2^{21} | + | | $2^{21}\d 3^9\d 5^4\d 7^2\d 11\d 13\d 23$ |
|- | |- | ||
| $Co_2$ | | $Co_2$ | ||
| − | | $2^{18} | + | | $2^{18}\d 3^6\d 5^3\d 7\d 11\d 23$ |
|- | |- | ||
| $Co_3$ | | $Co_3$ | ||
| − | | $2^{10} | + | | $2^{10}\d 3^7\d 5^3\d 7\d 11\d 23$ |
|- | |- | ||
| $F_{22}$, $M(22)$ | | $F_{22}$, $M(22)$ | ||
| rowspan="3" | Fischer groups | | rowspan="3" | Fischer groups | ||
| − | | $2^{17} | + | | $2^{17}\d 3^9\d 5^2\d 7\d 11\d 13$ |
|- | |- | ||
| $F_{23}$, $M(23)$ | | $F_{23}$, $M(23)$ | ||
| − | | $2^{18} | + | | $2^{18}\d 3^{13}\d 5^2\d 7\d 11\d 13\d 17\d 23$ |
|- | |- | ||
| $F_{24}^\prime$, $M(24)^\prime$ | | $F_{24}^\prime$, $M(24)^\prime$ | ||
| − | | $2^{21} | + | | $2^{21}\d 3^{16}\d 5^2\d 7^3\d 11\d 13\d 17\d 23\d 29$ |
|- | |- | ||
| $HS$ | | $HS$ | ||
| Higman–Sims group | | Higman–Sims group | ||
| − | | $2^9 | + | | $2^9\d 3^2\d 5^3\d 7\d 11$ |
|- | |- | ||
| $He$, $HHM$ | | $He$, $HHM$ | ||
| Held–Higman–McKay group | | Held–Higman–McKay group | ||
| − | | $2^{10} | + | | $2^{10}\d 3^3\d 5^2\d 7^3\d 17$ |
|- | |- | ||
| $Suz$ | | $Suz$ | ||
| Suzuki group | | Suzuki group | ||
| − | | $2^{13} | + | | $2^{13}\d 3^7\d 5^2\d 7\d 11\d 13$ |
|- | |- | ||
| $M^c$ | | $M^c$ | ||
| McLaughlin group | | McLaughlin group | ||
| − | | $2^7 | + | | $2^7\d 3^6\d 5^3\d 7\d 11$ |
|- | |- | ||
| $Ly$ | | $Ly$ | ||
| Lyons group | | Lyons group | ||
| − | | $2^8 | + | | $2^8\d 3^7\d 5^6\d 7\d 11\d 31\d 37\d 67$ |
|- | |- | ||
| $Ru$ | | $Ru$ | ||
| Rudvalis group | | Rudvalis group | ||
| − | | $2^{14} | + | | $2^{14}\d 3^3\d 5^3\d 7\d 13\d 29$ |
|- | |- | ||
| $O'N$, $O'NS$ | | $O'N$, $O'NS$ | ||
| O'Nan–Sims group | | O'Nan–Sims group | ||
| − | | $2^9 | + | | $2^9\d 3^4\d 5\d 7^3\d 11\d 19\d 31$ |
|- | |- | ||
| $F_1$, $M$ | | $F_1$, $M$ | ||
| Monster, Fischer–Griess group | | Monster, Fischer–Griess group | ||
| − | | $2^{46} | + | | $2^{46}\d 3^{20}\d 5^9\d 7^6\d 11^2\d 13^3\d 17\d 19\d 23\d 29\d 31\d 41\d 47\d 59\d 71$ |
|- | |- | ||
| $F_2$, $B$ | | $F_2$, $B$ | ||
| Baby monster | | Baby monster | ||
| − | | $2^{41} | + | | $2^{41}\d 3^{13}\d 5^6\d 7^2\d 11\d 13\d 17\d 19\d 23\d 31\d 47$ |
|- | |- | ||
| $F_3$, $E$, $Th$ | | $F_3$, $E$, $Th$ | ||
| Thompson group | | Thompson group | ||
| − | | $2^{15} | + | | $2^{15}\d 3^{10}\d 5^3\d 7^2\d 13\d 19\d 31$ |
|- | |- | ||
| $F_5$, $D$, $HN$ | | $F_5$, $D$, $HN$ | ||
| Harada–Norton group | | Harada–Norton group | ||
| − | | $2^{14} | + | | $2^{14}\d 3^6\d 5^6\d 7\d 11\d 19$ |
|} | |} | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | The | + | The classification of the finite simple groups (cf. {{Cite|As}}, {{Cite|Go}}) has led to the conclusion that |
| + | 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. | ||
| + | A discussion of the proof is given in {{Cite|Go}} up to the uniqueness | ||
| + | proof for the monster group $F_1$, which did appear in {{Cite|GrMeSe}}. | ||
| + | |||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | | valign="top"|{{Ref|As}}||valign="top"| M. Aschbacher, "The finite simple groups and their classification", Yale Univ. Press (1980) {{MR|0555880}} {{ZBL|0435.20007}} | ||
| + | |- | ||
| + | | valign="top"|{{Ref|CoCuNoPaWi}}||valign="top"| J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups", Clarendon Press (1985) {{MR|0827219}} {{ZBL|0568.20001}} | ||
| + | |- | ||
| + | | valign="top"|{{Ref|Go}}||valign="top"| D. Gorenstein, "Finite simple groups. An introduction to their classification", University Series in Mathematics. Plenum Publishing Corp., New York (1982) {{MR|0698782}} {{ZBL|0483.20008}} | ||
| + | |- | ||
| + | | valign="top"|{{Ref|GrMeSe}}||valign="top"| R.L. Griess, U. Meierfrankenfeld, Y. Segev, "A uniqueness proof for the Monster". ''Ann. of Math.'' (2) '''130''' (1989), no. 3, 567–602. | ||
| + | |- | ||
| + | | valign="top"|{{Ref|Sy}}||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 {{MR|0595144}} {{ZBL|0466.20006}} | ||
| + | |- | ||
| + | |} | ||
Revision as of 22:42, 30 April 2012
2020 Mathematics Subject Classification: Primary: 20D08 [MSN][ZBL]
A sporadic simple group is 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.
$\def\d{\cdot}$
| Notation | Name | Order |
|---|---|---|
| $M_{11}$ | Mathieu groups | $2^4\d 3^2\d 5\d 11$ |
| $M_{12}$ | $2^6\d 3^3\d 5\d 11$ | |
| $M_{22}$ | $2^7\d 3^2\d 5\d 7\d 11$ | |
| $M_{23}$ | $2^7\d 3^2\d 5\d 7\d 11\d 23$ | |
| $M_{24}$ | $2^{10}\d 3^3\d 5\d 7\d 11\d 23$ | |
| $J_1$ | Janko group | $2^3\d 3\d 5\d 7\d 11\d 19$ |
| $J_2$, $HJ$ | Hall–Janko group | $2^7\d 3^3\d 5^2\d 7$ |
| $J_3$, $HJM$ | Hall–Janko–McKay group | $2^7\d 3^5\d 5\d 17\d 19$ |
| $J_4$ | Janko group | $2^{21}\d 3^3\d 5\d 7\d 11^3\d 23\d 29\d 31\d 37\d 43$ |
| $Co_1$ | Conway groups | $2^{21}\d 3^9\d 5^4\d 7^2\d 11\d 13\d 23$ |
| $Co_2$ | $2^{18}\d 3^6\d 5^3\d 7\d 11\d 23$ | |
| $Co_3$ | $2^{10}\d 3^7\d 5^3\d 7\d 11\d 23$ | |
| $F_{22}$, $M(22)$ | Fischer groups | $2^{17}\d 3^9\d 5^2\d 7\d 11\d 13$ |
| $F_{23}$, $M(23)$ | $2^{18}\d 3^{13}\d 5^2\d 7\d 11\d 13\d 17\d 23$ | |
| $F_{24}^\prime$, $M(24)^\prime$ | $2^{21}\d 3^{16}\d 5^2\d 7^3\d 11\d 13\d 17\d 23\d 29$ | |
| $HS$ | Higman–Sims group | $2^9\d 3^2\d 5^3\d 7\d 11$ |
| $He$, $HHM$ | Held–Higman–McKay group | $2^{10}\d 3^3\d 5^2\d 7^3\d 17$ |
| $Suz$ | Suzuki group | $2^{13}\d 3^7\d 5^2\d 7\d 11\d 13$ |
| $M^c$ | McLaughlin group | $2^7\d 3^6\d 5^3\d 7\d 11$ |
| $Ly$ | Lyons group | $2^8\d 3^7\d 5^6\d 7\d 11\d 31\d 37\d 67$ |
| $Ru$ | Rudvalis group | $2^{14}\d 3^3\d 5^3\d 7\d 13\d 29$ |
| $O'N$, $O'NS$ | O'Nan–Sims group | $2^9\d 3^4\d 5\d 7^3\d 11\d 19\d 31$ |
| $F_1$, $M$ | Monster, Fischer–Griess group | $2^{46}\d 3^{20}\d 5^9\d 7^6\d 11^2\d 13^3\d 17\d 19\d 23\d 29\d 31\d 41\d 47\d 59\d 71$ |
| $F_2$, $B$ | Baby monster | $2^{41}\d 3^{13}\d 5^6\d 7^2\d 11\d 13\d 17\d 19\d 23\d 31\d 47$ |
| $F_3$, $E$, $Th$ | Thompson group | $2^{15}\d 3^{10}\d 5^3\d 7^2\d 13\d 19\d 31$ |
| $F_5$, $D$, $HN$ | Harada–Norton group | $2^{14}\d 3^6\d 5^6\d 7\d 11\d 19$ |
Comments
The classification of the finite simple groups (cf. [As], [Go]) has led to the conclusion that 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. A discussion of the proof is given in [Go] up to the uniqueness proof for the monster group $F_1$, which did appear in [GrMeSe].
References
| [As] | M. Aschbacher, "The finite simple groups and their classification", Yale Univ. Press (1980) MR0555880 Zbl 0435.20007 |
| [CoCuNoPaWi] | J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups", Clarendon Press (1985) MR0827219 Zbl 0568.20001 |
| [Go] | D. Gorenstein, "Finite simple groups. An introduction to their classification", University Series in Mathematics. Plenum Publishing Corp., New York (1982) MR0698782 Zbl 0483.20008 |
| [GrMeSe] | R.L. Griess, U. Meierfrankenfeld, Y. Segev, "A uniqueness proof for the Monster". Ann. of Math. (2) 130 (1989), no. 3, 567–602. |
| [Sy] | 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 MR0595144 Zbl 0466.20006 |
How to Cite This Entry:
Sporadic simple group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sporadic_simple_group&oldid=25784
Sporadic simple group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sporadic_simple_group&oldid=25784
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article