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\(nX\)-complementary generations of the sporadic group \(He\). (English) Zbl 1138.20015
Summary: A finite group \(G\) with conjugacy class \(nX\) is said to be \(nX\)-complementary generated if, given an arbitrary \(x\in G-\{1\}\), there is a \(y\in nX\) such that \(G=\langle x,y\rangle\). The \(nX\)-complementary generations of the simple groups were first introduced by A. J. Woldar [Commun. Algebra 22, No. 2, 675-685 (1994; Zbl 0806.20020)] to show that every sporadic simple group is \(pX\)-complementary generated for a suitable prime \(p\). In this paper we investigate all the \(nX\)-complementary generations of the Held group \(He\) and prove that the group \(He\) is \(nX\)-complementary generated if and only if \(nX=3B\) or \(n\geq 4\).
Reviewer: Reviewer (Berlin)

20D08 Simple groups: sporadic groups
20F05 Generators, relations, and presentations of groups