\beginexercise[Section 4.3, Exercise 15] Let $G$ be a $p$-group and let $N$ be a nontrivial normal subgroup of $G$. Prove that $N \cap Z(G) \neq 1$. \endexercise
\tableofcontents \newpage
\beginsolution Let $G$ act on $G/H = gH : g \in G$ by $g \cdot (xH) = (gx)H$. \beginenumerate \item \textbfTransitivity: Take any two cosets $aH, bH \in G/H$. Choose $g = ba^-1 \in G$. Then [ g \cdot (aH) = (ba^-1a)H = bH. ] Hence, there is exactly one orbit, so the action is transitive. \item \textbfStabilizer of $1H$: [ \Stab_G(1H) = g \in G : g \cdot (1H) = 1H = g \in G : gH = H. ] But $gH = H$ if and only if $g \in H$. Therefore $\Stab_G(1H) = H$. \endenumerate \endsolution
The exercises above illustrate the power of group actions in classifying finite groups, proving structural theorems (e.g., all groups of order $p^2$ are abelian), and laying the groundwork for the Sylow theorems. Mastery of Chapter 4 is essential for advanced topics such as representation theory, solvable groups, and the classification of finite simple groups.
Solutions Chapter 4 Overleaf: Dummit And Foote
\beginexercise[Section 4.3, Exercise 15] Let $G$ be a $p$-group and let $N$ be a nontrivial normal subgroup of $G$. Prove that $N \cap Z(G) \neq 1$. \endexercise
\tableofcontents \newpage
\beginsolution Let $G$ act on $G/H = gH : g \in G$ by $g \cdot (xH) = (gx)H$. \beginenumerate \item \textbfTransitivity: Take any two cosets $aH, bH \in G/H$. Choose $g = ba^-1 \in G$. Then [ g \cdot (aH) = (ba^-1a)H = bH. ] Hence, there is exactly one orbit, so the action is transitive. \item \textbfStabilizer of $1H$: [ \Stab_G(1H) = g \in G : g \cdot (1H) = 1H = g \in G : gH = H. ] But $gH = H$ if and only if $g \in H$. Therefore $\Stab_G(1H) = H$. \endenumerate \endsolution Dummit And Foote Solutions Chapter 4 Overleaf
The exercises above illustrate the power of group actions in classifying finite groups, proving structural theorems (e.g., all groups of order $p^2$ are abelian), and laying the groundwork for the Sylow theorems. Mastery of Chapter 4 is essential for advanced topics such as representation theory, solvable groups, and the classification of finite simple groups. \beginexercise[Section 4