Abstract
Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors \(v_1\) and \(v_2\) in V such that \(C_G(v_1) \cap C_G(v_2)=P\). A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Broué and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order.
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Gluck, D. Large \({\varvec{p}}^{\prime }\)-orbits for \({\varvec{p}}\)-nilpotent linear groups. Math. Z. 284, 1035–1052 (2016). https://doi.org/10.1007/s00209-016-1686-x
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DOI: https://doi.org/10.1007/s00209-016-1686-x