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| {{DISPLAYTITLE:Janko group J<sub>1</sub>}}
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| {{Group theory sidebar |Finite}}
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| In [[mathematics]], the smallest [[Janko group]], J<sub>1</sub>, is a simple [[sporadic group]] of order <math>175560=19 \cdot 11 \cdot 7 \cdot 5 \cdot 3 \cdot 2^3</math>. It was originally described by [[Zvonimir Janko]] (1965) and was the first sporadic group to be found since the discovery of the [[Mathieu group]]s in the 19th century. Its discovery launched the modern theory of [[sporadic group]]s.
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| == Properties ==
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| J<sub>1</sub> can be characterized abstractly as the unique [[simple group]] with abelian [[Sylow theorems|2-Sylow]] subgroups and with an [[Involution (mathematics)|involution]] whose [[centralizer]] is isomorphic to the [[direct product of groups|direct product]] of the group of order two and the [[alternating group]] A<sub>5</sup> of order 60, which is to say, the [[Icosahedral symmetry|rotational icosahedral group]]. That was Janko's original conception of the group.
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| In fact Janko and [[John G. Thompson|Thompson]] were investigating groups similar to the [[Ree group]]s <sup>2</sup>''G''<sub>2</sub>(3<sup>2''n''+1</sup>), and showed that if a simple group ''G'' has abelian Sylow 2-subgroups and a centralizer of an involution of the form '''Z'''/2'''Z'''×''PSL''<sub>2</sub>(''q'') for ''q'' a prime power at least 3, then either
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| ''q'' is a power of 3 and ''G'' has the same order as a Ree group (it was later shown that ''G'' must be a Ree group in this case) or ''q'' is 4 or 5. Note that ''PSL''<sub>2</sub>(''4'')=''PSL''<sub>2</sub>(''5'')=''A''<sub>5</sub>. This last exceptional case led to the Janko group J<sub>1</sub>.
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| J<sub>1</sub> has no [[outer automorphism group|outer automorphisms]] and its [[Schur multiplier]] is trivial.
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| J<sub>1</sub> is the smallest of the 6 sporadic simple groups called the [[pariah group|pariahs]], because they are not found within the [[Monster group]]. J<sub>1</sub> is contained in the [[O'Nan group]] as the subgroup of elements fixed by an outer automorphism of order 2.
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| ==Construction==
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| Janko found a [[modular representation]] in terms of 7 × 7 [[orthogonal matrix|orthogonal matrices]] in the [[finite field|field of eleven elements]], with generators given by
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| :<math>{\mathbf Y} = \left ( \begin{matrix}
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| 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
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| 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
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| 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right )</math>
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| and
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| :<math>{\mathbf Z} = \left ( \begin{matrix}
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| -3 & +2 & -1 & -1 & -3 & -1 & -3 \\
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| -2 & +1 & +1 & +3 & +1 & +3 & +3 \\
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| -1 & -1 & -3 & -1 & -3 & -3 & +2 \\
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| -1 & -3 & -1 & -3 & -3 & +2 & -1 \\
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| -3 & -1 & -3 & -3 & +2 & -1 & -1 \\
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| +1 & +3 & +3 & -2 & +1 & +1 & +3 \\
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| +3 & +3 & -2 & +1 & +1 & +3 & +1 \end{matrix} \right ).</math>
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| '''Y''' has order 7 and '''Z''' has order 5. Janko (1966) credited W. A. Coppel for recognizing this representation as an embedding into [[Leonard Eugene Dickson|Dickson's]] simple group [[Group of Lie type|''G''<sub>2</sub>(11)]] (which has a 7 dimensional representation over the field with 11 elements).
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| There is also a pair of generators a, b such that
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| :a<sup>2</sup>=b<sup>3</sup>=(ab)<sup>7</sup>=(abab<sup>−1</sup>)<sup>10</sup>=1
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| J<sub>1</sub> is thus a [[Hurwitz group]], a finite homomorphic image of the [[(2,3,7) triangle group]].
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| ==Maximal subgroups==
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| Janko (1966) enumerated all 7 conjugacy classes of maximal subgroups (see also the Atlas webpages cited below). Maximal simple subgroups of order 660 afford J<sub>1</sub> a [[permutation representation]] of degree 266. He found that there are 2 conjugacy classes of subgroups isomorphic to the [[alternating group]] A<sub>5</sub>, both found in the simple subgroups of order 660. J<sub>1</sub> has non-abelian simple proper subgroups of only 2 isomorphism types.
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| Here is a complete list of the maximal subgroups.
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| {| class="wikitable" style="margin: 1em auto; text-align: center;"
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| |-
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| ! Structure
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| ! Order
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| ! Index
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| ! Description
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| |-
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| | PSL<sub>2</sub>(11)
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| | 660
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| | 266
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| | Fixes point in smallest permutation representation
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| |-
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| | 2<sup>3</sup>.7.3
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| | 168
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| | 1045
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| | Normalizer of Sylow 2-subgroup
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| |-
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| | 2×A<sub>5</sub>
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| | 120
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| | 1463
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| | Centralizer of involution
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| |-
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| | 19.6
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| | 114
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| | 1540
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| | Normalizer of Sylow 19-subgroup
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| |-
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| | 11.10
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| | 110
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| | 1596
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| | Normalizer of Sylow 11-subgroup
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| |-
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| | D<sub>6</sub>×D<sub>10</sub>
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| | 60
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| | 2926
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| | Normalizer of Sylow 3-subgroup and Sylow 5-subgroup
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| |-
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| | 7.6
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| | 42
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| | 4180
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| | Normalizer of Sylow 7-subgroup
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| |}
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| The notation ''A''.''B'' means a group with a normal subgroup ''A'' with quotient ''B'', and
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| ''D''<sub>2''n''</sub> is the dihedral group of order 2''n''.
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| ==Number of elements of each order==
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| The greatest order of any element of the group is 19. The conjugacy class orders and sizes are found in the ATLAS.
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| {| class="wikitable" style="margin: 1em auto;"
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| |-
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| ! Order
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| ! No. elements
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| ! Conjugacy
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| |-
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| | 1 = 1
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| | 1 = 1
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| | 1 class
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| |-
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| | 2 = 2
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| | 1463 = 7 · 11 · 19
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| | 1 class
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| |-
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| | 3 = 3
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| | 5852 = 2<sup>2</sup> · 7 · 11 · 19
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| | 1 class
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| |-
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| | 5 = 5
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| | 11704 = 2<sup>3</sup> · 7 · 11 · 19
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| | 2 classes, power equivalent
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| |-
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| | 6 = 2 · 3
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| | 29260 = 2<sup>2</sup> · 5 · 7 · 11 · 19
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| | 1 class
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| |-
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| | 7 = 7
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| | 25080 = 2<sup>3</sup> · 3 · 5 · 11 · 19
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| | 1 class
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| |-
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| | 10 = 2 · 5
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| | 35112 = 2<sup>3</sup> · 3 · 7 · 11 · 19
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| | 2 classes, power equivalent
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| |-
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| | 11 = 11
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| | 15960 = 2<sup>3</sup> · 3 · 5 · 7 · 19
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| | 1 class
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| |-
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| | 15 = 3 · 5
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| | 23408 = 2<sup>4</sup> · 7 · 11 · 19
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| | 2 classes, power equivalent
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| |-
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| | 19 = 19
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| | 27720 = 2<sup>3</sup> · 3<sup>2</sup> · 5 · 7 · 11
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| | 3 classes, power equivalent
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| |}
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| == References ==
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| *{{Citation | last1=Chevalley | first1=Claude | title=Séminaire Bourbaki, Vol. 10 | origyear=1967 | url=http://www.numdam.org/item?id=SB_1966-1968__10__293_0 | publisher=[[Société Mathématique de France]] | location=Paris | mr=1610425 | year=1995 | chapter=Le groupe de Janko | pages=293–307}}
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| * Zvonimir Janko, ''A new finite simple group with abelian Sylow subgroups'', Proc. Nat. Acad. Sci. USA 53 (1965) 657-658.
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| * Zvonimir Janko, ''A new finite simple group with abelian Sylow subgroups and its characterization'', Journal of Algebra 3: 147-186, (1966) {{DOI|10.1016/0021-8693(66)90010-X}}
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| * Zvonimir Janko and John G. Thompson, ''On a Class of Finite Simple Groups of Ree'', Journal of Algebra, 4 (1966), 274-292.
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| * Robert A. Wilson, ''Is J<sub>1</sub> a subgroup of the monster?'', Bull. London Math. Soc. 18, no. 4 (1986), 349-350.
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| * [http://web.mat.bham.ac.uk/atlas/v2.0/spor/J1/ Atlas of Finite Group Representations: ''J''<sub>1</sub>] version 2
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J1/ Atlas of Finite Group Representations: ''J''<sub>1</sub>] version 3
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| [[Category:Sporadic groups]]
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Friends call him Royal. His day job is a cashier and his salary has been truly satisfying. To play croquet is some thing that I've carried out for many years. I presently live in Arizona but now I'm contemplating other choices.
Have a look at my page extended car warranty