|
|
| Line 1: |
Line 1: |
| {{Group theory sidebar}}
| | Environmental Consultant Cameron from Vegreville, has pastimes which include leathercrafting, property developers in [http://win-prizes-money.com/blogs/viewstory/54161 singapore property index] and base jumping. Recollects what a remarkable spot it was having paid a visit to Tabriz Historic Bazaar Complex. |
| | |
| In the [[mathematical]] field of [[group theory]], the '''Held group''' ''He'', found by {{harvs|first=Dieter|last=Held|year=1969a|year2=1969b|txt}}, is one of the 26 [[Sporadic group|sporadic]] [[simple group]]s. It is of the [[Order (group theory)|order]]
| |
| | |
| : 2<sup>10</sup>{{·}}3<sup>3</sup>{{·}}5<sup>2</sup>{{·}}7<sup>3</sup>{{·}}17
| |
| : = 4030387200
| |
| : ≈ 4{{·}}10<sup>9</sup>.
| |
| | |
| The Held group has [[Schur multiplier]] of order 1 and [[outer automorphism]] group of order 2.
| |
| | |
| == History ==
| |
| | |
| The group was found by Held 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 [[John McKay (mathematician)|John McKay]] and [[Graham Higman]].
| |
| | |
| == 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 automorphism group He.2 of the Held group He is a subgroup of the [[Fischer group Fi24]].
| |
| | |
| ==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 = [a,\, b^3]^5 = [a,\,babab^{-1}abab] =</math>
| |
| :<math>(ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1.</math>
| |
| | |
| ==Maximal subgroups==
| |
| {{harvtxt|Butler|1981}} found the 11 classes of maximal subgroups of the Held group 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 | url=http://dx.doi.org/10.1016/0021-8693(81)90127-7 | doi=10.1016/0021-8693(81)90127-7 | id={{MR|613857}} | year=1981 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=69 | issue=1 | pages=67–81}}
| |
| *{{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 | pages=253–296}}
| |
| *{{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}}
| |
| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/ Atlas of Finite Group Representations: Held group]
| |
| | |
| ==External links==
| |
| * [http://mathworld.wolfram.com/HeldGroup.html MathWorld: Held Group]
| |
| [[Category:Sporadic groups]]
| |
Environmental Consultant Cameron from Vegreville, has pastimes which include leathercrafting, property developers in singapore property index and base jumping. Recollects what a remarkable spot it was having paid a visit to Tabriz Historic Bazaar Complex.