Held group

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. MathSciNet|id=0249500DOI|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. MathSciNet|id=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]