\subsection*Exercise 17 Show that a group of order $p^2$ ($p$ prime) is abelian.
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Happy typesetting, and may your orbits be transitive and your Sylow subgroups conjugate. dummit+and+foote+solutions+chapter+4+overleaf+full
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The full solutions to Chapter 4 of Dummit and Foote on Overleaf can be accessed here: \subsection*Exercise 17 Show that a group of order
\beginproof Define $\psi: G/G_a \to \mathcalO_a$ by $\psi(gG_a)=g\cdot a$. Well-defined: $gG_a = hG_a \iff h^-1g\in G_a \iff (h^-1g)\cdot a = a \iff g\cdot a = h\cdot a$. $\psi$ is bijective (surjective by definition, injective by the previous equivalence). Hence $|\mathcalO_a| = |G/G_a| = [G:G_a]$. \endproof
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\beginproof $g\in \operatornameStab(H) \iff gHg^-1=H \iff g\in N_G(H)$. \endproof