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12.25 (1992)

Solved

Let $G$ be a finite group acting irreducibly on a vector space $V$. An orbit $\alpha^G$ for $\alpha \in V$ is said to be $p$-regular if the stabilizer of $\alpha$ in $G$ is a $p'$-subgroup. Does $G$ have a regular orbit on $V$ if it has a $p$-regular orbit for every prime $p$?

Contributor: Jiping Zhang

Progress

No, not always. For example, let $H = A_4 \wr \mathbb{Z}_5$, $V = O_2(H)$, and $G$ be a complement to $V$ in $H$, $G$ acting on $V$ by conjugation. (V. I. Zenkov, Letter of February, 11, 1994.)

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