Demonstrate that a given subset is not a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.2 Show that in each of the following examples, the specified subset is not a subgroup. (1)…
Surjective group endomorphisms need not be automorphisms
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.19 Let $G = \{ z \in \mathbb{C} \ |\ z^n = 1\ \mathrm{for\ some}\ n \in…
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…
Z/(n) is not a group under multiplication
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.5 Prove for all $n > 1$ that $\mathbb{Z}/(n)$ is not a group under multiplication of residue…
Decide whether or not a given binary operator is commutative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.2 Determine which of the following binary operations are commutative. The operation $\star$ on $\mathbb{Z}$ defined by…
Decide whether or not a given binary operator is associative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.1 Determine which of the following binary operations are associative. The operation $\star$ on $\mathbb{Z}$ defined by…