In the area of modern algebra known as
group theory, the Janko group ''J
3'' or the Higman-Janko-McKay group ''HJM'' is a
sporadic simple group
In 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 ...
of
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
: 2
73
551719 = 50232960.
History and properties
''J
3'' is one of the 26
Sporadic groups and was predicted by
Zvonimir Janko
Zvonimir Janko (26 July 1932 – 12 April 2022) was a Croatian mathematician who was the eponym of the Janko groups, sporadic simple groups in group theory. The first few sporadic simple groups were discovered by Émile Léonard Mathieu, which ...
in 1969 as one of two new simple groups having 2
1+4:A
5 as a centralizer of an involution (the other is the Janko group
''J2'').
''J
3'' was shown to exist by .
In 1982
R. L. Griess showed that ''J
3'' cannot be a
subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...
of the
monster group. Thus it is one of the 6 sporadic groups called the
pariahs
Pariah may refer to:
* A member of the Paraiyar caste in the Indian state of Tamil Nadu
* Pariah state, a country whose behavior does not conform to norms
* Outcast (person)
Science and mathematics
* Pariah dog, a type of semi-feral dog
* ''Pa ...
.
J
3 has an
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
of order 2 and a
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \oper ...
of order 3, and its triple cover has a unitary 9-dimensional
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
over the
finite field with 4 elements. 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.
Presentations
In terms of generators a, b, c, and d its automorphism group J
3:2 can be presented as
A presentation for J
3 in terms of (different) generators a, b, c, d is
Constructions
J3 can be constructed by many different
generators.
ATLAS page on J3
/ref> Two from the ATLAS list are 18x18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:
and
Maximal subgroups
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
*
* R. L. Griess, Jr., ''The Friendly Giant'', Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J3 is a pariah.
*
*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.
*
External links
MathWorld: Janko Groups
Atlas of Finite Group Representations: ''J''3
version 2
Atlas of Finite Group Representations: ''J''3
version 3
{{DEFAULTSORT:Janko group J3
Sporadic groups