In the area of modern algebra known as
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the Janko group ''J
3'' or the Higman-Janko-McKay group ''HJM'' is a
sporadic simple group of
order
: 2
73
551719 = 50232960.
History and properties
''J
3'' is one of the 26
Sporadic 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 ...
s and was predicted by
Zvonimir Janko 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 of the
monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
24632059761121331719232931414759 ...
. Thus it is one of the 6 sporadic groups called the
pariahs.
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 over the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with 4 elements. constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of order 9, with matrix multiplication carried out with finite field arithmetic In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers.
There are infin ...
:
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