View source for Held group

{{Short description|A sporadic simple group of order}}
{{Group theory sidebar |Finite}}

In the area of modern algebra known as [[Group theory|group theory]], the '''Held group''' ''He'' is a sporadic simple group of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;2<sup>10</sup>{{·}}3<sup>3</sup>{{·}}5<sup>2</sup>{{·}}7<sup>3</sup>{{·}}17  = 4030387200
: ≈ 4{{e|9}}.

==History==
''He'' is one of the 26 sporadic groups and was found by {{harvs|first=Dieter|last=Held|year=1969a|year2=1969b|txt}} during an investigation of simple groups containing an involution whose centralizer is isomorphic to that of an involution in the [[Mathieu group M24|Mathieu group M<sub>24</sub>]].  A second such group is the [[Projective linear group|linear group]] L<sub>5</sub>(2). The Held group is the third possibility, and its construction was completed by [[Biography:John McKay (mathematician)|John McKay]] and [[Biography:Graham Higman|Graham Higman]].

The [[Outer automorphism group|outer automorphism group]] has order 2 and the [[Schur multiplier]] 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]]. 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|vertex operator algebra]] over the field with 7 elements.

The smallest permutation representation is a rank 5 action on 2058 points with point stabilizer Sp<sub>4</sub>(4):2. 

The automorphism group He:2 of the Held group He is a subgroup of the [[Fischer group Fi24|Fischer group Fi<sub>24</sub>]].

==Generalized monstrous moonshine==

Conway and Norton suggested in their 1979 paper that [[Monstrous moonshine|monstrous moonshine]] 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 <math>T_{7A}(\tau)</math> where one can set the constant term a(0) = 10 ({{OEIS2C|id=A007264}}),

:<math>\begin{align}
     j_{7A}(\tau)
  &= T_{7A}(\tau)+10\\
  &= \left(\left(\tfrac{\eta(\tau)}{\eta(7\tau)}\right)^{2} + 7\left(\tfrac{\eta(7\tau)}{\eta(\tau)}\right)^2\right)^2\\
  &= \frac{1}{q} + 10 + 51q + 204q^2 + 681q^3 + 1956q^4 + 5135q^5 + \dots
\end{align}</math>

and ''η''(''τ'') is the [[Dedekind eta function]].

==Presentation==
It can be defined in terms of the generators ''a'' and ''b'' and relations
:<math>a^2 = b^7 = (ab)^{17} = [a, b]^6 = \left [a, b^3 \right ]^5 = \left [a, babab^{-1}abab \right ] = (ab)^4 ab^2 ab^{-3} ababab^{-1}ab^3 ab^{-2}ab^2 = 1.</math>

==Maximal subgroups==
{{harvtxt|Butler|1981}} found the 11 conjugacy classes of maximal subgroups of ''He'' as follows:

* S<SUB>4</SUB>(4):2
* 2<SUP>2</SUP>.L<SUB>3</SUB>(4).S<SUB>3</SUB>
* 2<SUP>6</SUP>:3.S<SUB>6</SUB>
* 2<SUP>6</SUP>:3.S<SUB>6</SUB>
* 2<SUP>1+6</SUP>.L<SUB>3</SUB>(2)
* 7<SUP>2</SUP>:2.L<SUB>2</SUB>(7)
* 3.S<SUB>7</SUB>
* 7<SUP>1+2</SUP>:(3 × S<SUB>3</SUB>)
* S<SUB>4</SUB> × L<SUB>3</SUB>(2)
* 7:3 × L<SUB>3</SUB>(2)
* 5<SUP>2</SUP>:4A<SUB>4</SUB>

==References==
*{{Citation | last1=Butler | first1=Gregory | title=The maximal subgroups of the sporadic simple group of Held | doi=10.1016/0021-8693(81)90127-7 |mr=613857 | year=1981 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=69 | issue=1 | pages=67–81| doi-access=free }}
*{{citation|first=D.|last=Held|contribution=Some simple groups related to ''M''<sub>24</sub>|editor1-first=Richard|editor1-last=Brauer|editor2-first=Chih-Han|editor2-last=Shah|title=Theory of Finite Groups: A Symposium|publisher=W. A. Benjamin|year=1969a}}
*{{Citation | last1=Held | first1=Dieter | title=The simple groups related to ''M''<sub>24</sub>| doi=10.1016/0021-8693(69)90074-X | mr=0249500 | year=1969b | journal=[[Journal of Algebra]] | volume=13 | issue=2 | pages=253–296| doi-access=free }}
*{{citation|last=Ryba|first=A. J. E.|title=Calculation of the 7-modular characters of the Held group | journal=[[Journal of Algebra]]|volume=  117|year=1988|issue= 1|pages= 240–255|mr=0955602|doi=10.1016/0021-8693(88)90252-9|doi-access=}} 

==External links==
* [http://mathworld.wolfram.com/HeldGroup.html MathWorld: Held group]
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/ Atlas of Finite Group Representations: Held group]\

[[Category:Sporadic groups]]

{{Sourceattribution|Held group}}