Dummit And Foote Solutions Chapter 4 Overleaf Now

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Alternatively, consider the action of $G$ on the set of all subsets of size $n$? A standard proof uses the regular representation and the sign homomorphism. Let $G$ act on itself by left multiplication; this yields an embedding $\pi: G \hookrightarrow S_2n$. Since $n$ is odd, $2n$ is even. Compose with the sign map $\sgn: S_2n \to \pm1$. The kernel of $\sgn \circ \pi$ is a subgroup of index at most $2$. If the image is $\pm1$, the kernel has index $2$ and hence order $n$. If the image is trivial, then every element acts as an even permutation. But in $S_2n$, a transposition is odd; careful analysis (see D&F) shows this forces a contradiction for $n$ odd. Thus the kernel is the desired subgroup of order $n$. \endsolution

\beginexercise[Section 4.2, Exercise 2] Let $G$ act on a finite set $A$. Prove that if $G$ acts transitively on $A$, then $|A|$ divides $|G|$. \endexercise Dummit And Foote Solutions Chapter 4 Overleaf

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\beginsolution Let $n_3$ denote the number of Sylow $3$-subgroups. By Sylow's theorems, $n_3 \equiv 1 \pmod3$ and $n_3 \mid 4$. The divisors of $4$ are $1,2,4$. Which are $\equiv 1 \pmod3$? $1 \equiv 1 \pmod3$, $4 \equiv 1 \pmod3$, but $2 \equiv 2 \pmod3$. Hence $n_3 = 1$ or $n_3 = 4$. No other possibilities. \endsolution

\documentclass[12pt, leqno]article \usepackage[utf8]inputenc \usepackageamsmath, amssymb, amsthm, amscd \usepackage[margin=1in]geometry \usepackageenumitem \usepackagetitlesec \usepackagexcolor % -------------------------------------------------------------- % Custom Commands for Dummit & Foote Notation % -------------------------------------------------------------- \newcommand\Z\mathbbZ \newcommand\R\mathbbR \newcommand\C\mathbbC \newcommand\Q\mathbbQ \newcommand\F\mathbbF \newcommand\Stab\textStab \newcommand\Fix\textFix \newcommand\Orb\textOrb \newcommand\sgn\textsgn \newcommand\Aut\textAut \newcommand\Inn\textInn \newcommand\soc\textSoc \newcommand\Ker\textKer \newcommand\Image\textIm \begindocument Alternatively, consider the action of $G$ on

\sectionApplications to $p$-groups and Sylow Theorems

% Custom colors for clarity \definecolornoteRGB0,100,0 Since $n$ is odd, $2n$ is even

% -------------------------------------------------------------- % Title & Author % -------------------------------------------------------------- \titleSolutions to Dummit & Foote\ Chapter 4: Group Actions \authorPrepared for Overleaf \date\today

% Theorem environments \newtheoremtheoremTheorem[section] \newtheoremlemma[theorem]Lemma \newtheoremproposition[theorem]Proposition \newtheoremcorollary[theorem]Corollary \theoremstyledefinition \newtheoremdefinition[theorem]Definition \newtheoremexample[theorem]Example \newtheoremexerciseExercise[section] \newtheoremsolutionSolution[section]