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{{Short description|Sporadic simple group}}
{{Group theory sidebar |Finite}}

In the area of modern algebra known as [[group theory]], the '''Held group''' ''He'' is a [[sporadic simple group]] of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;4,030,387,200 = 2<sup>10</sup>{{·}}3<sup>3</sup>{{·}}5<sup>2</sup>{{·}}7<sup>3</sup>{{·}}17
: ≈ 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 an extension of the [[extra special group]] 2<sup>1+6</sup> by the [[projective linear group|linear group]] L<sub>3</sub>(2), which is the same involution centralizer as the [[Mathieu group M24|Mathieu group M<sub>24</sub>]].  A second such group is the linear group L<sub>5</sub>(2). The Held group is the third possibility, and its construction was completed by [[John McKay (mathematician)|John McKay]] and [[Graham Higman]]. In all of these groups, the extension splits.

The [[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 [[centralizer|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]] 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 [[Graph (discrete mathematics)|graph]] associated with this representation has rank 5 and is [[Directed graph|directed]]; the outer automorphism reverses the direction of the edges, decreasing the rank to 4.

Since He is the normalizer of a [[Frobenius group]] 7:3 in the [[Monster group]], it does not just commute with a 7-cycle, but also some 3-cycles. Each of these 3-cycles is normalized by the [[Fischer group Fi24|Fischer group Fi<sub>24</sub>]], so He:2 is a subgroup of the derived subgroup Fi<sub>24</sub>' (the non-simple group Fi<sub>24</sub> has 2 conjugacy classes of He:2, which are fused by an outer automorphism). As mentioned above, the smallest permutation representation of He has 2058 points, and when realized inside Fi<sub>24</sub>', there is an [[Group action#Orbits and stabilizers|orbit]] of 2058 [[Fischer groups|transpositions]].

==Generalized monstrous moonshine==

Conway and Norton suggested in their 1979 paper that [[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:
{| class="wikitable"
|+ Maximal subgroups of ''He''
|-
! No. !! Structure !! Order !! Index !! Comments
|-
|  1||S<sub>4</sub>(4):2                                      ||align=right|1,958,400<br />=&nbsp;2<sup>9</sup>·3<sup>2</sup>·5<sup>2</sup>·17||align=right|    2,058<br />=&nbsp;2·3·7<sup>3</sup>                             ||
|-
|  2||2<sup>2</sup>.L<sub>3</sub>(4).S<sub>3</sub>            ||align=right|  483,840<br />=&nbsp;2<sup>9</sup>·3<sup>3</sup>·5·7             ||align=right|    8,330<br />=&nbsp;2·5·7<sup>2</sup>·17                          ||
|-
|3,4||2<sup>6</sup>:3<sup>&thinsp;·&thinsp;</sup>S<sub>6</sub>||align=right|  138,240<br />=&nbsp;2<sup>10</sup>·3<sup>3</sup>·5              ||align=right|   29,155<br />=&nbsp;5·7<sup>3</sup>·17                            || two classes, fused by an outer automorphism
|-
|  5||2{{su|a=l|b=+|p=1+6}}:L<sub>3</sub>(2)                  ||align=right|   21,504<br />=&nbsp;2<sup>10</sup>·3·7                          ||align=right|  187,425<br />=&nbsp;3<sup>2</sup>·5<sup>2</sup>·7<sup>2</sup>·17  || centralizer of an involution of class 2B
|-
|  6||7<sup>2</sup>:2.L<sub>2</sub>(7)                        ||align=right|   16,464<br />=&nbsp;2<sup>4</sup>·3·7<sup>3</sup>               ||align=right|  244,800<br />=&nbsp;2<sup>6</sup>·3<sup>2</sup>·5<sup>2</sup>·17  ||
|-
|  7||3.S<sub>7</sub>                                         ||align=right|   15,120<br />=&nbsp;2<sup>4</sup>·3<sup>3</sup>·5·7             ||align=right|  266,560<br />=&nbsp;2<sup>6</sup>·5·7<sup>2</sup>·17              || normalizer of a subgroup of order 3 (class 3A); centralizer of an outer automorphism of order 2
|-
|  8||7{{su|a=l|b=+|p=1+2}}:(3 × S<sub>3</sub>)               ||align=right|    6,174<br />=&nbsp;2·3<sup>2</sup>·7<sup>3</sup>               ||align=right|  652,800<br />=&nbsp;2<sup>9</sup>·3·5<sup>2</sup>·17              || normalizer of a subgroup of order 7 (class 7C)
|-
|  9||S<sub>4</sub> × L<sub>3</sub>(2)                        ||align=right|    4,032<br />=&nbsp;2<sup>6</sup>·3<sup>2</sup>·7               ||align=right|  999,600<br />=&nbsp;2<sup>4</sup>·3·5<sup>2</sup>·7<sup>2</sup>·17||
|-
| 10||7:3 × L<sub>3</sub>(2)                                  ||align=right|    3,528<br />=&nbsp;2<sup>3</sup>·3<sup>2</sup>·7<sup>2</sup>   ||align=right|1,142,400<br />=&nbsp;2<sup>7</sup>·3·5<sup>2</sup>·7·17            ||
|-
| 11||5<sup>2</sup>:4A<sub>4</sub>                            ||align=right|    1,200<br />=&nbsp;2<sup>4</sup>·3·5<sup>2</sup>               ||align=right|3,358,656<br />=&nbsp;2<sup>6</sup>·3<sup>2</sup>·7<sup>3</sup>·17  ||
|}

==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]]