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 Held group ''He'' is a
sporadic simple group of
order
: 2
103
35
27
317 = 4030387200
: ≈ 4.
History
''He'' is one of the 26 sporadic groups and was found by during an investigation of simple groups containing an involution whose centralizer is isomorphic to that of an involution in the
Mathieu group M24. A second such group is the
linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a f ...
L
5(2). The Held group is the third possibility, and its construction was completed by
John McKay and
Graham Higman
Graham Higman FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory.
Biography
Higman was born in Louth, Lincolnshire, and attended Sutton High School, Plymouth, winning a ...
.
The
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 ...
has order 2 and the
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 ...
is trivial.
Representations
The smallest faithful complex representation has dimension 51; there are two such representations that are duals of each other.
It
centralizes an element of order 7 in 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
246320597611213317192329314147 ...
. As a result the prime 7 plays a special role in the theory of the group; for example, the smallest representation of the Held group over any field is the 50-dimensional representation over the field with 7 elements, and it acts naturally on a
vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
over the field with 7 elements.
The smallest permutation representation is a rank 5 action on 2058 points with point stabilizer Sp
4(4):2.
The automorphism group He:2 of the Held group He is a subgroup of the
Fischer group Fi24.
Generalized monstrous moonshine
Conway and Norton suggested in their 1979 paper that
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For ''He'', the relevant McKay-Thompson series is
where one can set the constant term a(0) = 10 (),
:
and ''η''(''τ'') is the
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
.
Presentation
It can be defined in terms of the generators ''a'' and ''b'' and relations
:
Maximal subgroups
found the 11 conjugacy classes of maximal subgroups of ''He'' as follows:
* S
4(4):2
* 2
2.L
3(4).S
3
* 2
6:3.S
6
* 2
6:3.S
6
* 2
1+6.L
3(2)
* 7
2:2.L
2(7)
* 3.S
7
* 7
1+2:(3 × S
3)
* S
4 × L
3(2)
* 7:3 × L
3(2)
* 5
2:4A
4
References
*
*
*
*{{citation, last=Ryba, first=A. J. E., title=Calculation of the 7-modular characters of the Held group , journal=
Journal of Algebra
''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1 ...
, volume= 117, year=1988, issue= 1, pages= 240–255, mr=0955602, doi=10.1016/0021-8693(88)90252-9, doi-access=free
External links
MathWorld: Held groupAtlas of Finite Group Representations: Held group
Sporadic groups