The kernel and stabilizers of a group action are subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.4 Let $G$ be a group acting on a set $A$. Show that the following sets are…
Examine a given action of the additive real numbers on the Cartesian plane
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.3 Show that the additive group $\mathbb{R}$ acts on the $xy$-plane $\mathbb{R} \times \mathbb{R}$ by $r \cdot…
The additive group of integers acts on itself by left addition
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.2 Show that the additive group $\mathbb{Z}$ acts on itself by $z \cdot a = z+a$ for…
The group of units of a field acts on the field by left multiplication
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.1 Solution: Let $a \in F$. We have $1 \cdot a = 1a = a$ by the…
Compute the stabilizer of an element under a given action of Sym(n)
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 $A$…
Exhibit an action of Sym(4) on a set of polynomials
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…
Sym(4) acts on a set of polynomials by permuting the variables
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,…