Held group

Group theory

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In the mathematical field of group theory, the Held group He (found by Dieter Held (1969)) is one of the 26 sporadic simple groups, and has order

2¹⁰ · 3³ · 5² · 7³ · 17

= 4030387200

≈ 4 · 10⁹.

It can be defined in terms of the generators a and b and relations

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

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

It 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.

The Held group has Schur multiplier of order 1 and outer automorphism group of order 2.

It centralizes an element of order 7 in the Monster group (but is not a subgroup of any of the Conway groups). 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.

References

D. Held Some simple groups related to M₂₄, in Richard Brauer and Chih-Han Shah, "Theory of Finite Groups: A Symposium", W. A. Benjamin (1969)
Held, Dieter The simple groups related to M₂₄ J. Algebra 13 1969 253-296. MR 0249500 doi:10.1016/0021-8693(69)90074-X
Ryba, A. J. E. Calculation of the 7-modular characters of the Held group. J. Algebra 117 (1988), no. 1, 240--255. MR 0955602 doi:10.1016/0021-8693(88)90252-9
Atlas of Finite Group Representations: Held group

External links

MathWorld: Held Group