Dummit+and+foote+solutions+chapter+4+overleaf+full ((full))

\sectionGroup Actions and Permutation Representations

\subsection*Exercise 14 Let $|G|=pq$ with primes $p<q$ and $p \nmid q-1$. Show $G$ is cyclic. dummit+and+foote+solutions+chapter+4+overleaf+full

\beginproof The center of $G$, denoted $Z(G)$, is non-trivial for any $p$-group. Thus $|Z(G)|$ is either $p$ or $p^2$. \beginenumerate \item Suppose $|Z(G)| = p^2$. Then $Z(G) = G$, so $G$ is abelian. \item Suppose $|Z(G)| = p$. Then the order of the quotient $G/Z(G)$ is $p$. Groups of prime order are cyclic. Let $G/Z(G) = \langle xZ(G) \rangle$. dummit+and+foote+solutions+chapter+4+overleaf+full

Linking the size of orbits and stabilizers, foundational for counting arguments. dummit+and+foote+solutions+chapter+4+overleaf+full

\titleDummit & Foote Chapter 4 Solutions: Group Actions \authorYour Name \date\today