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\beginexercise[Section 4.5, Exercise 3] Let $G$ be a finite group, $p$ a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Prove that $N_G(N_G(P)) = N_G(P)$. \endexercise Dummit And Foote Solutions Chapter 4 Overleaf
\beginexercise[Section 4.2, Exercise 8] Let $G$ be a $p$-group acting on a finite set $A$. Prove that [ |A| \equiv |\Fix(A)| \pmodp, ] where $\Fix(A) = a \in A : g \cdot a = a \text for all g \in G$. \endexercise \tableofcontents \newpage \beginexercise[Section 4
% -------------------------------------------------------------- % Title & Author % -------------------------------------------------------------- \titleSolutions to Dummit & Foote\ Chapter 4: Group Actions \authorPrepared for Overleaf \date\today \tableofcontents \newpage \beginexercise[Section 4.5
\beginthebibliography9 \bibitemDF Dummit, David S., and Richard M. Foote. \textitAbstract Algebra. 3rd ed., Wiley, 2004. \endthebibliography