sporadic

• adj.

• 散発的な、時々起こる、散発性の、散在する、散在性の、散在的な
• 孤発性の

　 　

WordNet

1. recurring in scattered and irregular or unpredictable instances; "a city subjected to sporadic bombing raids"
2. in a sporadic manner; "he only works sporadically" (同)periodically

PrepTutorEJDIC

1. ときどき起こる; / 散在する / (病気が伝染性でなく)散発の,特発性の
2. 時折;まばらに;散発的に

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/01/23 01:37:09」(JST)

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In group theory, a discipline within mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.

A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families, plus 26 exceptions that do not follow such a systematic pattern. These are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case the sporadic groups number 27.

The monster group is the largest of the sporadic groups and contains all but six of the other sporadic groups as subgroups or subquotients.

Names of the sporadic groups

Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is:

The diagram shows the subquotient relations between the sporadic groups.

• Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄
• Janko groups J₁, J₂ or HJ, J₃ or HJM, J₄
• Conway groups Co₁, Co₂, Co₃
• Fischer groups Fi₂₂, Fi₂₃, Fi₂₄′ or F₃₊
• Higman–Sims group HS
• McLaughlin group McL
• Held group He or F₇₊ or F₇
• Rudvalis group Ru
• Suzuki group Suz or F₃₋
• O'Nan group O'N
• Harada–Norton group HN or F₅₊ or F₅
• Lyons group Ly
• Thompson group Th or F_3|3 or F₃
• Baby Monster group B or F₂₊ or F₂
• Fischer–Griess Monster group M or F₁

The Tits group T is sometimes also regarded as a sporadic group (it is almost but not strictly a group of Lie type), which is why in some sources the number of sporadic groups is given as 27 instead of 26. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type.

Matrix representations over finite fields for all the sporadic groups have been constructed.

The earliest use of the term "sporadic group" may be Burnside (1911, p. 504, note N) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received".

The diagram on the right is based on the diagram given in Ronan (2006). The sporadic groups also have a lot of subgroups which are not sporadic but these are not shown on the diagram because they are too numerous.

Organization

Of the 26 sporadic groups, 20 can be seen inside the Monster group as subgroups or quotients of subgroups (sections).

I. Pariah

Main article: Pariah group

The six exceptions are J₁, J₃, J₄, O'N, Ru and Ly. These six are sometimes known as the pariahs.

II. Happy Family

The remaining twenty have been called the Happy Family by Robert Griess, and can be organized into three generations.

First generation (5 groups): the Mathieu groups

Main article: Mathieu groups

M_n for n = 11, 12, 22, 23 and 24 are multiply transitive permutation groups on n points. They are all subgroups of M₂₄, which is a permutation group on 24 points.

Second generation (7 groups): the Leech lattice

See also: Leech lattice and Conway groups

All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:

• Co₁ is the quotient of the automorphism group by its center {±1}
• Co₂ is the stabilizer of a type 2 (i.e., length 2) vector
• Co₃ is the stabilizer of a type 3 (i.e., length √6) vector
• Suz is the group of automorphisms preserving a complex structure (modulo its center)
• McL is the stabilizer of a type 2-2-3 triangle
• HS is the stabilizer of a type 2-3-3 triangle
• J₂ is the group of automorphisms preserving a quaternionic structure (modulo its center).

Third generation (8 groups): other subgroups of the Monster

Consists of subgroups which are closely related to the Monster group M:

• B or F₂ has a double cover which is the centralizer of an element of order 2 in M
• Fi₂₄′ has a triple cover which is the centralizer of an element of order 3 in M (in conjugacy class "3A")
• Fi₂₃ is a subgroup of Fi₂₄′
• Fi₂₂ has a double cover which is a subgroup of Fi₂₃
• The product of Th = F₃ and a group of order 3 is the centralizer of an element of order 3 in M (in conjugacy class "3C")
• The product of HN = F₅ and a group of order 5 is the centralizer of an element of order 5 in M
• The product of He = F₇ and a group of order 7 is the centralizer of an element of order 7 in M.
• Finally, the Monster group itself is considered to be in this generation.

(This series continues further: the product of M₁₂ and a group of order 11 is the centralizer of an element of order 11 in M.)

The Tits group also belongs in this generation: there is a subgroup S₄ ײF₄(2)′ normalising a 2C² subgroup of B, giving rise to a subgroup 2·S₄ ײF₄(2)′ normalising a certain Q₈ subgroup of the Monster. ²F₄(2)′ is also a subgroup of the Fischer groups Fi₂₂, Fi₂₃ and Fi₂₄′, and of the Baby Monster B. ²F₄(2)′ is also a subgroup of the (pariah) Rudvalis group Ru, and has no involvements in sporadic simple groups except the containments we have already mentioned.

Table of the sporadic group orders

Group	Generation	Order (sequence A001228 in OEIS)	1SF	Factorized order
F1 or M	third	8080174247945128758864599049617107 57005754368000000000	≈ 8×1053	246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
F2 or B	third	4154781481226426191177580544000000	≈ 4×1033	241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47
Fi24' or F3+	third	1255205709190661721292800	≈ 1×1024	221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29
Fi23	third	4089470473293004800	≈ 4×1018	218 · 313 · 52 · 7 · 11 · 13 · 17 · 23
Fi22	third	64561751654400	≈ 6×1013	217 · 39 · 52 · 7 · 11 · 13
F3 or Th	third	90745943887872000	≈ 9×1016	215 · 310 · 53 · 72 · 13 · 19 · 31
Ly	pariah	51765179004000000	≈ 5×1016	28 · 37 · 56 · 7 · 11 · 31 · 37 · 67
F5 or HN	third	273030912000000	≈ 3×1014	214 · 36 · 56 · 7 · 11 · 19
Co1	second	4157776806543360000	≈ 4×1018	221 · 39 · 54 · 72 · 11 · 13 · 23
Co2	second	42305421312000	≈ 4×1013	218 · 36 · 53 · 7 · 11 · 23
Co3	second	495766656000	≈ 5×1011	210 · 37 · 53 · 7 · 11 · 23
O'N	pariah	460815505920	≈ 5×1011	29 · 34 · 5 · 73 · 11 · 19 · 31
Suz	second	448345497600	≈ 4×1011	213 · 37 · 52 · 7 · 11 · 13
Ru	pariah	145926144000	≈ 1×1011	214 · 33 · 53 · 7 · 13 · 29
F7 or He	third	4030387200	≈ 4×109	210 · 33 · 52 · 73 · 17
McL	second	898128000	≈ 9×108	27 · 36 · 53 · 7 · 11
HS	second	44352000	≈ 4×107	29 · 32 · 53 · 7 · 11
J4	pariah	86775571046077562880	≈ 9×1019	221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43
J3 or HJM	pariah	50232960	≈ 5×107	27 · 35 · 5 · 17 · 19
J2 or HJ	second	604800	≈ 6×105	27 · 33 · 52 · 7
J1	pariah	175560	≈ 2×105	23 · 3 · 5 · 7 · 11 · 19
M24	first	244823040	≈ 2×108	210 · 33 · 5 · 7 · 11 · 23
M23	first	10200960	≈ 1×107	27 · 32 · 5 · 7 · 11 · 23
M22	first	443520	≈ 4×105	27 · 32 · 5 · 7 · 11
M12	first	95040	≈ 1×105	26 · 33 · 5 · 11
M11	first	7920	≈ 8×103	24 · 32 · 5 · 11

References

• Burnside, William (1911), Theory of groups of finite order, p. 504 (note N), ISBN 0-486-49575-2 
• Conway, J. H. (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups" (PDF), Proc. Nat. Acad. Sci. U.S.A. 61 (2): 398–400, doi:10.1073/pnas.61.2.398, Zbl 0186.32401 
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press. ISBN 0-19-853199-0. Zbl 0568.20001. 
• Gorenstein, D.; Lyons, R.; Solomon, R. (1994), The Classification of the Finite Simple Groups, American Mathematical Society  Issues 1, 2, ...
• Griess, Robert L. (1998), Twelve Sporadic Groups, Springer-Verlag, ISBN 3540627782, Zbl 0908.20007 
• Ronan, Mark (2006), Symmetry and the Monster, Oxford, ISBN 978-0-19-280722-9, Zbl 1113.00002 

External links

• Weisstein, Eric W., "Sporadic Group", MathWorld.
• Atlas of Finite Group Representations: Sporadic groups

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English Journal

• An update on the role of mitochondrial α-ketoglutarate dehydrogenase in oxidative stress.

• Starkov AA.SourceDepartment of Neurology and Neuroscience, Weill Medical College, Cornell University, New York, NY 10021, USA. Electronic address: ans2024@med.cornell.edu.
• Molecular and cellular neurosciences.Mol Cell Neurosci.2013 Jul;55:13-6. doi: 10.1016/j.mcn.2012.07.005. Epub 2012 Jul 20.
• The activity of mitochondrial alpha-ketoglutarate dehydrogenase complex (KGDHC) is severely reduced in human pathologies where oxidative stress is traditionally thought to play an important role, such as familial and sporadic forms of Alzheimer's disease and other age-related neurodegenerative disea
• PMID 22820180

• Mitochondria and ALS: Implications from novel genes and pathways.

• Cozzolino M, Ferri A, Valle C, Carrì MT.SourceInstitute for Translational Pharmacology, CNR, Rome, Italy; Fondazione Santa Lucia IRCCS, Rome, Italy.
• Molecular and cellular neurosciences.Mol Cell Neurosci.2013 Jul;55:44-9. doi: 10.1016/j.mcn.2012.06.001. Epub 2012 Jun 15.
• Evidence from patients with sporadic and familiar amyotrophic lateral sclerosis (ALS) and from models based on the overexpression of mutant SOD1 found in a small subset of patients, clearly point to mitochondrial damage as a relevant facet of this neurodegenerative condition. In this mini-review we
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• PFN1 mutations are rare in Han Chinese populations with amyotrophic lateral sclerosis.

• Chen Y, Zheng ZZ, Huang R, Chen K, Song W, Zhao B, Chen X, Yang Y, Yuan L, Shang HF.SourceDepartment of Neurology and State Key Laboratory of Biotherapy and Cancer Center, West China Hospital, SiChuan University, Chengdu, Sichuan, China.
• Neurobiology of aging.Neurobiol Aging.2013 Jul;34(7):1922.e1-5. doi: 10.1016/j.neurobiolaging.2013.01.013. Epub 2013 Feb 19.
• Amyotrophic lateral sclerosis (ALS) is a fatal neurodegenerative disease with unknown pathophysiological mechanisms. Profilin 1 gene (PFN1) has been identified as a causative gene, which accounts for 1% to 2% of familial ALS. In this study, we investigated the mutation spectrum of PFN1 in Chinese pa
• PMID 23428184

Japanese Journal

• Tumor necrosis factor (TNF) and chemokines in colitis-associated cancer

• Mukaida Naofumi,Sasakki So-ichiro,Popivanova Boryana K.
• Cancers 3(3), 2811-2826, 2011-09
• … Moreover, intratumoral infiltration of inflammatory cells suggests the involvement of inflammatory responses also in other forms of sporadic as well as heritable colon cancer. …
• NAID 120003479719

• The FOXE1 and NKX2-1 loci are associated with susceptibility to papillary thyroid carcinoma in the Japanese population.

• Matsuse Michiko,Takahashi Meiko,Mitsutake Norisato,Nishihara Eijun,Hirokawa Mitsuyoshi,Kawaguchi Takahisa,Rogounovitch Tatiana,Saenko Vladimir,Bychkov Andrey,Suzuki Keiji,Matsuo Keitaro,Tajima Kazuo,Miyauchi Akira,Yamada Ryo,Matsuda Fumihiko,Yamashita Shunichi
• Journal of Medical Genetics 48(9), 645-648, 2011-09
• … Background FOXE1 and NKX2-1 are two known genetic risk factors for the predisposition to sporadic papillary thyroid carcinoma (PTC) in Europeans, but their association in other ethnicities is still unknown. … Objective We aim to examine the association of the two genes with Japanese sporadic PTC, which exhibits high BRAF(V600E) mutation rate. … Methods 507 Japanese sporadic PTC cases and 2766 controls were genotyped for rs965513 (FOXE1) and rs944289 (NKX2-1). …
• NAID 120003307704

Related Links

• Sporadic - definition of sporadic by The Free Dictionary

spo·rad·ic (spə-răd′ĭk, spô-) also spo·rad·i·cal (-ĭ-kəl) adj. Occurring at irregular intervals or in isolated or scattered places; having no pattern or order: his sporadic attendance in class; rebels who put up sporadic resistance. See ...

• Sporadic | Definition of sporadic by Merriam-Webster

Full Definition of SPORADIC: occurring occasionally, singly, or in irregular or random instances <sporadic protests> <a sporadic disease> — spo · rad · i · cal · ly \-di-k(ə-)lē\ adverb See sporadic defined for English-language learners ...

• Sporadic | Define Sporadic at Dictionary.com

Sporadic definition, (of similar things or occurrences) appearing or happening at irregular intervals in time; occasional: sporadic renewals of enthusiasm. See more. Dictionary.com Word of the Day Translate Games Blog Apps ...

Related Pictures

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★リンクテーブル★

リンク元	「散発性」「dispersive」「孤発性」「孤発」「孤発型」
拡張検索	「sporadic retinoblastoma」「sporadically」「sporadic inclusion body myositis」「sporadic late-onset Alzheimer's disease」

「散発性」

　　[★]

英

sporadic、dispersive、sporadically

関

散発的、分散的、孤発性、孤発、孤発型

「dispersive」

　　[★]

• 散発性の、散発的な、分散的な

関

sporadic、sporadically

「孤発性」

　　[★]

英

sporadic、sporadically

関

散発性、孤発、孤発型

「孤発」

　　[★]

英

sporadic

関

散発性、孤発性、孤発型

「孤発型」

　　[★]

英

sporadic

関

散発性、孤発性、孤発

「sporadic retinoblastoma」

　　[★]

散発性網膜芽細胞種

関

familial retinoblastoma、hereditary retinoblastoma、retinoblastoma

「sporadically」

　　[★]

• adv.

• 孤発性に、散発性に

関

dispersive、sporadic

　 　

「sporadic inclusion body myositis」

　　[★]

孤発性封入体筋炎

関

inclusion body myositis

「sporadic late-onset Alzheimer's disease」

　　[★] 孤発性遅発性アルツハイマー病