Mathieu group M22
{{Short description|Sporadic simple group}}
{{for|general background and history of the Mathieu sporadic groups|Mathieu group}}
{{DISPLAYTITLE:Mathieu group M22}}
{{Group theory sidebar |Finite}}
In the area of modern algebra known as group theory, the Mathieu group M22 is a sporadic simple group of order
: 443,520 = 27{{·}}32{{·}}5{{·}}7{{·}}11
: ≈ 4{{e|5}}.
History and properties
M22 is one of the 26 sporadic groups and was introduced by {{harvs|txt |authorlink=Émile Léonard Mathieu |last=Mathieu |year1=1861 |year2=1873}}. It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M22 is cyclic of order 12, and the outer automorphism group has order 2.
There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature.
{{harvtxt|Burgoyne|Fong|1966}} incorrectly claimed that the Schur multiplier of M22 has order 3, and in a correction {{harvtxt|Burgoyne|Fong|1968}} incorrectly claimed that it has order 6. This caused an error in the title of the paper {{harvtxt|Janko|1976}} announcing the discovery of the Janko group J4. {{harvtxt|Mazet|1979}} showed that the Schur multiplier is in fact cyclic of order 12.
{{harvtxt|Adem|Milgram|1995}} calculated the 2-part of all the cohomology of M22.
Representations
M22 has a 3-transitive permutation representation on 22 points, with point stabilizer the group PSL3(4), sometimes called M21. This action fixes a Steiner system S(3,6,22) with 77 hexads, whose full automorphism group is the automorphism group M22.2 of M22.
M22 has three rank 3 permutation representations: one on the 77 hexads with point stabilizer 24:A6, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A7.
M22 is the point stabilizer of the action of M23 on 23 points, and also the point stabilizer of the rank 3 action of the Higman–Sims group on 100 = 1+22+77 points.
The triple cover 3.M22 has a 6-dimensional faithful representation over the field with 4 elements.
The 6-fold cover of M22 appears in the centralizer 21+12.3.(M22:2) of an involution of the Janko group J4.
Maximal subgroups
There are no proper subgroups transitive on all 22 points. There are 8 conjugacy classes of maximal subgroups of M22 as follows:
| class="wikitable"
|+ Maximal subgroups of M22 | ||||
| No. | Structure | Order | Index | Comments |
|---|---|---|---|---|
| 1 | M21 ≅ L3(4) | align=right|20,160 = 26·32·5·7 | align=right| 22 = 2·11 | one-point stabilizer |
| 2 | 24:A6 | align=right| 5,760 = 27·32·5 | align=right| 77 = 7·11 | has orbits of sizes 6 and 16; stabilizer of W22 block |
| 3,4 | A7 | align=right| 2,520 = 23·32·5·7 | align=right|176 = 24·11 | two classes, fused by an outer automorphism; has orbits of sizes 7 and 15; there are 2 sets, of 15 each, of simple subgroups of order 168. Those of one type have orbits of sizes 1, 7, and 14; the others have orbits of sizes 7, 8, and 7. |
| 5 | 24:S5 | align=right| 1,920 = 27·3·5 | align=right|231 = 3·7·11 | has orbits of sizes 2 and 20 (5 blocks of size 4); a 2-point stabilizer in the sextet group |
| 6 | 23:L3(2) | align=right| 1,344 = 26·3·7 | align=right|330 = 2·3·5·11 | has orbits of sizes 8 and 14; centralizer of an outer automorphism of order 2 (class 2B) |
| 7 | M10 ≅ A6·23 | align=right| 720 = 24·32·5 | align=right|616 = 23·7·11 | has orbits of sizes 10 and 12 (2 blocks of size 6); a one-point stabilizer of M11 (point in orbit of 11) |
| 8 | L2(11) | align=right| 660 = 22·3·5·11 | align=right|672 = 25·3·7 | has two orbits of size 11; another one-point stabilizer of M11 (point in orbit of 12) |
Conjugacy classes
There are 12 conjugacy classes, though the two classes of elements of order 11 are fused under an outer automorphism.
| class="wikitable" style="margin: 1em auto;" | |||
| Order | No. elements | Cycle structure | |
|---|---|---|---|
| 1 = 1 | 1 | 122 | |
| 2 = 2 | 1155 = 3 · 5 · 7 · 11 | 1628 | |
| 3 = 3 | 12320 = 25 · 5 · 7 · 11 | 1436 | |
| rowspan="2" | 4 = 22 | 13860 = 22 · 32 · 5 · 7 · 11 | 122244 | |
| |27720 = 23 · 32 · 5 · 7 · 11 | 122244 | ||
| 5 = 5 | 88704 = 27 · 32 · 7 · 11 | 1254 | |
| 6 = 2 · 3 | 36960 = 25 · 3 · 5 · 7 · 11 | 223262 | |
| rowspan="2" | 7 = 7 | 63360= 27 · 32 · 5 · 11 | 1 73 | rowspan="2" | Power equivalent |
| |63360= 27 · 32 · 5 · 11 | 1 73 | ||
| 8 = 23 | 55440 = 24 · 32 · 5 · 7 · 11 | 2·4·82 | |
| rowspan="2" | 11 = 11 | 40320 = 27 · 32 · 5 · 7 | 112 | rowspan="2" | Power equivalent |
| |40320 = 27 · 32 · 5 · 7 | 112 |
See also
References
- {{Citation | last1=Adem | first1=Alejandro |author1-link= Alejandro Adem | last2=Milgram | first2=R. James | title=The cohomology of the Mathieu group M₂₂ | doi=10.1016/0040-9383(94)00029-K |mr=1318884 | year=1995 | journal=Topology | issn=0040-9383 | volume=34 | issue=2 | pages=389–410| doi-access=free }}
- {{Citation | last1=Burgoyne | first1=N. | last2=Fong | first2=Paul | title=The Schur multipliers of the Mathieu groups | url=http://projecteuclid.org/euclid.nmj/1118801786 |mr=0197542 | year=1966 | journal=Nagoya Mathematical Journal | issn=0027-7630 | volume=27 | issue=2 | pages=733–745| doi=10.1017/S0027763000026519 | doi-access=free }}
- {{Citation | last1=Burgoyne | first1=N. | last2=Fong | first2=Paul | title=A correction to: "The Schur multipliers of the Mathieu groups" | url=http://projecteuclid.org/euclid.nmj/1118796952 |mr=0219626 | year=1968 | journal=Nagoya Mathematical Journal | issn=0027-7630 | volume=31 | pages=297–304| doi=10.1017/S0027763000012782 | doi-access=free }}
- {{Citation | last1=Cameron | first1=Peter J. | title=Permutation Groups | publisher=Cambridge University Press | series=London Mathematical Society Student Texts | isbn=978-0-521-65378-7 | year=1999 | volume=45 | url-access=registration | url=https://archive.org/details/permutationgroup0000came }}
- {{Citation | last1=Carmichael | first1=Robert D. | title=Introduction to the theory of groups of finite order | orig-year=1937 | url=https://books.google.com/books?id=McMgAAAAMAAJ | publisher=Dover Publications | location=New York | isbn=978-0-486-60300-1 |mr=0075938 | year=1956}}
- {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | editor1-last=Powell | editor1-first=M. B. | editor2-last=Higman | editor2-first=Graham | editor2-link=Graham Higman | title=Finite simple groups | chapter-url=https://books.google.com/books?id=TPPkAAAAIAAJ | publisher=Academic Press | location=Boston, MA | series=Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969. | isbn=978-0-12-563850-0 |mr=0338152 | year=1971 | chapter=Three lectures on exceptional groups | pages=215–247}} Reprinted in {{harvtxt|Conway|Sloane|1999|loc= 267–298}}
- {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Parker | first2=Richard A. | last3=Norton | first3=Simon P. | last4=Curtis | first4=R. T. | last5=Wilson | first5=Robert A.| title=Atlas of finite groups | url=https://books.google.com/books?id=38fEMl2-Fp8C | publisher=Oxford University Press | isbn=978-0-19-853199-9 |mr=827219 | year=1985}}
- {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=Neil J. A. | author2-link=Neil Sloane | title=Sphere Packings, Lattices and Groups | url=https://books.google.com/books?id=upYwZ6cQumoC | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-98585-5 |mr=0920369 | year=1999 | volume=290 | doi=10.1007/978-1-4757-2016-7}}
- {{Citation
|title=The Mathieu groups and their geometries
|first=Hans
|last=Cuypers
|url=http://www.win.tue.nl/~hansc/mathieu.pdf
}}
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- {{Citation | last1=Griess | first1=Robert L. Jr. | author1-link=R. L. Griess | title=Twelve sporadic groups | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-62778-4 |mr=1707296 | year=1998 | doi=10.1007/978-3-662-03516-0}}
- {{Citation | last1=Harada | first1=Koichiro | last2=Solomon | first2=Ronald | title=Finite groups having a standard component L of type M₁₂ or M₂₂ | doi=10.1016/j.jalgebra.2006.09.034 |mr=2381799 | year=2008 | journal=Journal of Algebra | issn=0021-8693 | volume=319 | issue=2 | pages=621–628| doi-access=free }}
- {{cite journal | last1 = Janko | first1 = Z. | year = 1976 | title = A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroups | journal = J. Algebra | volume = 42 | pages = 564–596 | doi = 10.1016/0021-8693(76)90115-0 | doi-access = free }} (The title of this paper is incorrect, as the full covering group of M22 was later discovered to be larger: center of order 12, not 6.)
- {{Citation | last1=Mathieu | first1=Émile | title=Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables | url=http://gallica.bnf.fr/ark:/12148/bpt6k16405f/f249 | year=1861 | journal=Journal de Mathématiques Pures et Appliquées | volume=6 | pages=241–323}}
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- {{Citation | last1=Mazet | first1=Pierre | title=Sur le multiplicateur de Schur du groupe de Mathieu M₂₂ |mr=560327 | year=1979 | journal=Comptes Rendus de l'Académie des Sciences, Série A et B | issn=0151-0509 | volume=289 | issue=14 | pages=A659–A661}}
- {{Citation | last1=Thompson | first1=Thomas M. | title=From error-correcting codes through sphere packings to simple groups | url=https://books.google.com/books?id=ggqxuG31B3cC | publisher=Mathematical Association of America | series=Carus Mathematical Monographs | isbn=978-0-88385-023-7 |mr=749038 | year=1983 | volume=21}}
- {{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=über Steinersche Systeme | doi=10.1007/BF02948948 | year=1938a | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | issn=0025-5858 | volume=12 | pages=265–275}}
- {{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=Die 5-fach transitiven Gruppen von Mathieu | doi=10.1007/BF02948947 | year=1938b | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume=12 | pages=256–264}}
External links
- [http://mathworld.wolfram.com/MathieuGroups.html MathWorld: Mathieu Groups]
- [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M22/ Atlas of Finite Group Representations: M22]
{{DEFAULTSORT:Mathieu Group M22}}