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Nilpotent residual of fixed points (1712.08103v1)
Published 21 Dec 2017 in math.GR
Abstract: Let $q$ be a prime and $A$ a finite $q$-group of exponent $q$ acting by automorphisms on a finite $q'$-group $G$. Assume that $A$ has order at least $q3$. We show that if $\gamma_{\infty} (C_{G}(a))$ has order at most $m$ for any $a \in A{#}$, then the order of $\gamma_{\infty} (G)$ is bounded solely in terms of $m$ and $q$. If $\gamma_{\infty} (C_{G}(a))$ has rank at most $r$ for any $a \in A{#}$, then the rank of $\gamma_{\infty} (G)$ is bounded solely in terms of $r$ and $q$.