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 $\mathbb{Z}/(3)[S_3]$: $$\alpha = 1(2\ 3) + 2(1\ 2\…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.10 Consider the following elements of the integral group ring $\mathbb{Z}[S_3]$: $$\alpha = 3(1\ 2) - 5(2\…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.16 Let $G$ be a group with $x,y \in G$. Assume $|x| = n$ and $|y| =…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.14 Let $\sigma = (1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12)$. For…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.13 Let $n$ be a positive integer and let $R$ be the set of all polynomials with…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.12 Let $R$ be the set of all polynomials with integers coefficients in the independent variables $x_1,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.8 Let $G = S_n$ and fix $i \in \{ 1, 2, \ldots, n \}$. Prove that…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.5 For each of the following subgroups $A$ of a given group $G$, show that $C_G(A) =…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.4 For each of the groups $S_3$, $D_8$, and $Q_8$, compute the centralizer of each element and…