## Compute the orbits, cycle decompositions, and stabilizers of some given group actions of Sym(3)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.4 Let $S_3$ act on the set $A = \{ (i,j) \ |\ 1 \leq i,j \leq…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.4 Let $S_3$ act on the set $A = \{ (i,j) \ |\ 1 \leq i,j \leq…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.11 Write out the cycle decomposition of the eight permutations in $S_4$ corresponding to the elements of…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.10 Let $k \in \mathbb{Z}^+$. (1) For which values of $k$ is the action of $S_A$ on…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.9 Let $A$ be a nonempty set and let $k \in \mathbb{Z}^+$ such that $k \leq |A|$.…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.8 Let $A$ be a nonempty set and let $k \in \mathbb{Z}^+$ such that $k \leq |A|$.…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.15 Fix $i \in \{ 1, \ldots, n \} = A$, and let $S_n$ act on $A$…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.14 Prove that $S_4$ does not have a normal subgroup of order 8 or a normal subgroup…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.13 Fix any labeling of the vertices of a square and use this to identify $D_8$ as…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.2 Find the lattice of subgroups of $S_3$. Solution: Every (nontrivial) $\subseteq$-minimal subgroup of $S_3$ is cyclic…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.11 Consider the following elements of the group ring $\mathbb{Z}/(3)[S_3]$: $$\alpha = 1(2\ 3) + 2(1\ 2\…