Held group

In the mathematical field of group theory, the Held group He, found by Dieter Held (1969a, 1969b), is one of the 26 sporadic simple groups. It is of the order

2¹⁰ · 3³ · 5² · 7³ · 17

= 4030387200

≈ 4 · 10⁹.

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. A second such group is the linear group L₅(2). The Held group is the third possibility, and its construction was completed by 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 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 SP₄(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

$a^{2}=b^{7}=(ab)^{{17}}=[a,\,b]^{6}=[a,\,b^{3}]^{5}=[a,\,babab^{{-1}}abab]=$

$(ab)^{4}ab^{2}ab^{{-3}}ababab^{{-1}}ab^{3}ab^{{-2}}ab^{2}=1.$

Maximal subgroups

Butler (1981) found the 11 classes of maximal subgroups of the Held group as follows.

S₄(4):2

2².L₃(4).S₃

2⁶:3.S₆

2¹⁺⁶.L₃(2)

7²:2.L₂(7)

3.S₇

7¹⁺²:(3 × S₃)

S₄ × L₃(2)

7:3 × L₃(2)

5²:4A₄

References

Butler, Gregory (1981), "The maximal subgroups of the sporadic simple group of Held", Journal of Algebra 69 (1): 67–81, doi:10.1016/0021-8693(81)90127-7, ISSN 0021-8693, MR 613857
Held, D. (1969a), "Some simple groups related to M₂₄", in Brauer, Richard; Shah, Chih-Han, Theory of Finite Groups: A Symposium, W. A. Benjamin
Held, Dieter (1969b), "The simple groups related to M₂₄", Journal of Algebra 13: 253–296, doi:10.1016/0021-8693(69)90074-X, MR 0249500
Ryba, A. J. E. (1988), "Calculation of the 7-modular characters of the Held group", Journal of Algebra 117 (1): 240–255, doi:10.1016/0021-8693(88)90252-9, MR 0955602
Atlas of Finite Group Representations: Held group

External links

MathWorld: Held Group

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