Held group

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In the mathematical field of group theory, the Held group He (named after Dieter Held) is a finite group of order

2¹⁰ · 3³ · 5² · 7³ · 17

= 4030387200

≈ 4 · 10⁹.

It is a simple group, meaning it does not have any normal subgroups except for the subgroup consisting only of the identity element, and He itself.

The Held group is one of the 26 sporadic groups, and is the smallest of the "third generation" of these groups. It can be defined in terms of the generators a and b and relations

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

(a b) 4 a b 2 a b - 3 a b a b a b - 1 a b 3 a b - 2 a b 2 = 1.

It is 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.

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