Sporadic simple group
For general background and history of the Janko sporadic groups, see
Janko group .
In the area of modern algebra known as group theory , the Janko group J3 or the Higman-Janko-McKay group HJM is a sporadic simple group of order
27 · 35 · 5 · 17 · 19 = 50232960.
History and properties
J3 is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 21+4 :A5 as a centralizer of an involution (the other is the Janko group J2 J3 was shown to exist by Graham Higman and John McKay (1969 ).
In 1982 R. L. Griess showed that J3 cannot be a subquotient of the monster group .[ 1] pariahs .
J3 has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9-dimensional representation over the finite field with 4 elements. Weiss (1982) constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the finite field with 9 elements.
It has a complex projective representation of dimension eighteen.
Constructions
Using matrices
J3 can be constructed by many different generators .[ 2] finite field of order 9, with matrix multiplication carried out with finite field arithmetic :
(
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
3
7
4
8
4
8
1
5
5
1
2
0
8
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
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0
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0
0
0
0
0
8
0
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0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
4
8
6
2
4
8
0
4
0
8
4
5
0
8
1
1
8
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
)
{\displaystyle \left({\begin{matrix}0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0\\3&7&4&8&4&8&1&5&5&1&2&0&8&6&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8\\4&8&6&2&4&8&0&4&0&8&4&5&0&8&1&1&8&5\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&0\\\end{matrix}}\right)}
and
(
4
8
0
0
0
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0
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0
0
0
0
0
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0
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0
8
0
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4
4
8
0
0
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8
0
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8
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0
8
0
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8
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8
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8
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2
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0
0
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1
1
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8
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0
8
8
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0
0
8
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8
2
5
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7
2
8
1
5
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7
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0
0
7
3
8
)
{\displaystyle \left({\begin{matrix}4&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\4&4&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0\\0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0\\0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0\\2&7&4&5&7&4&8&5&6&7&2&2&8&8&0&0&5&0\\4&7&5&8&6&1&1&6&5&3&8&7&5&0&8&8&6&0\\0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0\\8&2&5&5&7&2&8&1&5&5&7&8&6&0&0&7&3&8\\\end{matrix}}\right)}
The automorphism group J 3 :2 can be constructed by starting with the subgroup PSL(2,16):4 and adjoining 120 involutions, which are identified with the Sylow 17-subgroups. Note that these 120 involutions are outer elements of J 3 :2. One then defines the following relation:
(
1
1
1
0
σ σ -->
t
(
ν ν -->
,
ν ν -->
7
)
)
5
=
1
{\displaystyle \left({\begin{matrix}1&1\\1&0\end{matrix}}\sigma t_{(\nu ,\nu 7)}\right)^{5}=1}
where
σ σ -->
{\displaystyle \sigma }
Frobenius automorphism or order 4, and
t
(
ν ν -->
,
ν ν -->
7
)
{\displaystyle t_{(\nu ,\nu 7)}}
∞ ∞ -->
→ → -->
0
→ → -->
1
→ → -->
7
{\displaystyle \infty \rightarrow 0\rightarrow 1\rightarrow 7}
Curtis showed, using a computer, that this relation is sufficient to define J 3 :2.[ 3]
Using a presentation
In terms of generators a, b, c, and d its automorphism group J3 :2 can be presented as
a
17
=
b
8
=
a
b
a
− − -->
2
=
c
2
=
b
c
b
3
=
(
a
b
c
)
4
=
(
a
c
)
17
=
d
2
=
[
d
,
a
]
=
[
d
,
b
]
=
(
a
3
b
− − -->
3
c
d
)
5
=
1.
{\displaystyle a^{17}=b^{8}=a^{b}a^{-2}=c^{2}=b^{c}b^{3}=(abc)^{4}=(ac)^{17}=d^{2}=[d,a]=[d,b]=(a^{3}b^{-3}cd)^{5}=1.}
A presentation for J3 in terms of (different) generators a, b, c, d is
a
19
=
b
9
=
a
b
a
2
=
c
2
=
d
2
=
(
b
c
)
2
=
(
b
d
)
2
=
(
a
c
)
3
=
(
a
d
)
3
=
(
a
2
c
a
− − -->
3
d
)
3
=
1.
{\displaystyle a^{19}=b^{9}=a^{b}a^{2}=c^{2}=d^{2}=(bc)^{2}=(bd)^{2}=(ac)^{3}=(ad)^{3}=(a^{2}ca^{-3}d)^{3}=1.}
Maximal subgroups
Finkelstein & Rudvalis (1974) found the 9 conjugacy classes of maximal subgroups of J3 as follows:
PSL(2,16):2, order 8160
PSL(2,19), order 3420
PSL(2,19), conjugate to preceding class in J3 :2
24 : (3 × A5 ), order 2880
PSL(2,17), order 2448
(3 × A6 ):22 , order 2160 - normalizer of subgroup of order 3
32+1+2 :8, order 1944 - normalizer of Sylow 3-subgroup
21+4 :A5 , order 1920 - centralizer of involution
22+4 : (3 × S3 ), order 1152
References
^ Griess (1982): p. 93: proof that J3 is a pariah.
^ ATLAS page on J3 ^ Bradley, J.D.; Curtis, R.T. (2006), "Symmetric Generationand existence of J 3 :2, the automorphism group of the third Janko group", Journal of Algebra , 304 (1): 256–270, doi :10.1016/j.jalgebra.2005.09.046
Finkelstein, L.; Rudvalis, A. (1974), "The maximal subgroups of Janko's simple group of order 50,232,960", Journal of Algebra 30 (1–3): 122–143, doi :10.1016/0021-8693(74)90196-3 ISSN 0021-8693 , MR 0354846 R. L. Griess , Jr., The Friendly Giant , Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J3 is a pariah.Higman, Graham ; McKay, John (1969), "On Janko's simple group of order 50,232,960", Bull. London Math. Soc. , 1 : 89–94, correction p. 219, doi :10.1112/blms/1.1.89 , MR 0246955 Z. Janko, Some new finite simple groups of finite order , 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25–64 Academic Press, London, and in The theory of finite groups (Edited by Brauer and Sah) p. 63-64, Benjamin, 1969.MR 0244371
Weiss, Richard (1982). "A Geometric Construction of Janko's Group J3 ". Mathematische Zeitschrift . 179 (179): 91–95. doi :10.1007/BF01173917 .
External links
![{\displaystyle a^{17}=b^{8}=a^{b}a^{-2}=c^{2}=b^{c}b^{3}=(abc)^{4}=(ac)^{17}=d^{2}=[d,a]=[d,b]=(a^{3}b^{-3}cd)^{5}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58d1b22df8960eecb0fa0cb95e013899c64d4e30)
