Home  

Random  

Nearby  



Log in  



Settings  



Donate  



About Wikipedia  

Disclaimers  



Wikipedia





Janko group J4





Article  

Talk  



Language  

Watch  

Edit  





In the area of modern algebra known as group theory, the Janko group J4 is a sporadic simple groupoforder

   86,775,571,046,077,562,880
= 221 ·33 ···113 · 23 · 29 · 31 · 37 ·43
≈ 9×1019.

History

edit

J4 is one of the 26 Sporadic groups. Zvonimir Janko found J4 in 1975 by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field with 2 elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. Aschbacher & Segev (1991) and Ivanov (1992) gave computer-free proofs of uniqueness. Ivanov & Meierfrankenfeld (1999) and Ivanov (2004) gave a computer-free proof of existence by constructing it as an amalgams of groups 210:SL5(2) and (210:24:A8):2 over a group 210:24:A8.

The Schur multiplier and the outer automorphism group are both trivial.

Since 37 and 43 are not supersingular primes, J4 cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

Representations

edit

The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements.

The smallest permutation representation is on 173067389 points and has rank 20, with point stabilizer of the form 211:M24. The points can be identified with certain "special vectors" in the 112 dimensional representation.

Presentation

edit

It has a presentation in terms of three generators a, b, and c as

 

Alternatively, one can start with the subgroup M24 and adjoin 3975 involutions, which are identified with the trios. By adding a certain relation, certain products of commuting involutions generate the binary Golay cocode, which extends to the maximal subgroup 211:M24. Bolt, Bray, and Curtis showed, using a computer, that adding just one more relation is sufficient to define J4.

Maximal subgroups

edit

Kleidman & Wilson (1988) found the 13 conjugacy classes of maximal subgroups of J4 which are listed in the table below.

Maximal subgroups of J4
No. Structure Order Comments
1 211:M24 501,397,585,920
= 221·33·5·7·11·23
contains a Sylow 2-subgroup and a Sylow 3-subgroup; contains the centralizer 211:(M22:2) of involution of class 2B
2 21+12
+
·3.(M22:2)
21,799,895,040
= 221·33·5·7·11
centralizer of involution of class 2A; contains a Sylow 2-subgroup and a Sylow 3-subgroup
3 210:L5(2) 10,239,344,640
= 220·32·5·7·31
4 23+12·(S5 × L3(2)) 660,602,880
= 221·32·5·7
contains a Sylow 2-subgroup
5 U3(11):2 141,831,360
= 26·32·5·113·37
6 M22:2 887,040
= 28·32·5·7·11
7 111+2
+
:(5 × GL(2,3))
319,440
= 24·3·5·113
normalizer of a Sylow 11-subgroup
8 L2(32):5 163,680
= 25·3·5·11·31
9 PGL(2,23) 12,144
= 24·3·11·23
10 U3(3) 6,048
= 25·33·7
contains a Sylow 3-subgroup
11 29:28 812
= 22·7·29
Frobenius group; normalizer of a Sylow 29-subgroup
12 43:14 602
= 2·7·43
Frobenius group; normalizer of a Sylow 43-subgroup
13 37:12 444
= 22·3·37
Frobenius group; normalizer of a Sylow 37-subgroup

A Sylow 3-subgroup of J4 is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3.

References

edit
edit

Retrieved from "https://en.wikipedia.org/w/index.php?title=Janko_group_J4&oldid=1230011165"
 



Last edited on 20 June 2024, at 01:37  





Languages

 



This page is not available in other languages.
 

Wikipedia


This page was last edited on 20 June 2024, at 01:37 (UTC).

Content is available under CC BY-SA 4.0 unless otherwise noted.



Privacy policy

About Wikipedia

Disclaimers

Contact Wikipedia

Code of Conduct

Developers

Statistics

Cookie statement

Terms of Use

Desktop