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 large powers modulo n
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.23 Determine the last two digits of $3^{3^{100}}$. (Find $3^{100} \mod {\varphi(100)}$ and use Exercise 3.2.22.) Solution:…
Use Lagrange’s Theorem to prove Euler’s Theorem
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.22 Use Lagrange’s Theorem in the multiplicative group $G = (\mathbb{Z}/(n))^\times$ to prove Euler’s Theorem: if $\mathsf{gcd}(a,n)…
The additive group of rational numbers has no subgroups of finite index
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.21 Prove that $\mathbb{Q}$ has no proper subgroups of finite index. Deduce that $\mathbb{Q}/\mathbb{Z}$ has no proper…
The intersection by an abelian normal subgroup is normal in the product
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.20 Let $G$ be a group and $A,B \leq G$ be subgroups such that $A$ is abelian…
Normal subgroups whose order and index are coprime are unique up to order
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.19 Let $G$ be a finite group, $N \leq G$ a normal subgroup, and suppose that $|N|$…
If the index and order of a normal subgroup and subgroup are relatively prime, then the subgroup is contained in the normal subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.18 Let $G$ be a group and let $H,N \leq G$ with $N$ normal in $G$. Prove…