Compute the order of a cyclic subgroup in Z/(54)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.10 What is the order of $\overline{30}$ in $\mathbb{Z}/(54)$? Write out all of the elements and their…
If the group and an element have the same finite order then the group is cyclic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.2 Let $G$ be a finite group and let $x \in G$. Prove that if $|x| =…
The order of a general linear group over a finite field is bounded above
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.6 Let $F$ be a field. If $|F| = q$ is finite show that $|GL_n(F)| < q^{n^2}$.…
Compute the order of each element in the general linear group of dimension 2 over Z/(2)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.2 Write out all the elements in $GL_2(\mathbb{F}_2)$ and compute the order of each element. Solution: We…
Set of all elements of a given order is not necessarily a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.14 Let $n \geq 3$. Show that $\{ x \in D_{2n} \ |\ x^2 = 1\}$ is…
If a finite group has a generating set containing two elements of order 2, then it is a dihedral group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.24 Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order…
The multiplicative groups of nonzero real numbers and nonzero complex numbers are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.4 Prove that the multiplicative groups $\mathbb{R}^\times$ and $\mathbb{C}^\times$ are not isomorphic. Solution: Recall from Exercise 1.6.2…
Group isomorphisms preserve the order of an element
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.2 Let $G$ and $H$ be groups. If $\varphi : G \rightarrow H$ is an isomorphism, show…
There is a unique noncyclic group of order 4
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.36 Assume that $G = \{1, a, b, c\}$ is a group of order 4 with identity…
Characterize the elements of cyclic subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.35 Let $G$ be a group, and let $x \in G$ be an element of finite order;…