A solvable version of the Baer–Suzuki theorem

Guest, Simon

doi:10.1090/S0002-9947-2010-04932-3

A solvable version of the Baer–Suzuki theorem
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Trans. Amer. Math. Soc. 362 (2010), 5909-5946 Request permission

Abstract:

Suppose that $G$ is a finite group and $x \in G$ has prime order $p \ge 5$. Then $x$ is contained in the solvable radical of $G$, $O_{\infty }(G)$, if (and only if) $\langle x,x^g \rangle$ is solvable for all $g \in G$. If $G$ is an almost simple group and $x \in G$ has prime order $p \ge 5$, then this implies that there exists $g \in G$ such that $\langle x,x^g \rangle$ is not solvable. In fact, this is also true when $p=3$ with very few exceptions, which are described explicitly.

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