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…
Group homomorphisms preserve exponents
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.1 Let $G$ and $H$ be groups and $\varphi : G \rightarrow H$ a group homomorphism. (1)…
Detecting copies of Dih(2n) in a larger group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.2 Exercise 1.2.6 Let $G$ be a group, and let $x,y \in G$ be elements of order 2. Prove…
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;…
The order of a group element is smaller than the cardinality of the group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.34 Let $G$ be a group and $x \in G$ an element of order $n < \infty$.…
The powers of a group element of infinite order are pairwise distinct
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.33 Let $G$ be a group, and let $x \in G$ be an element of infinite order.…
Properties of inverses in a finite cyclic subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.32 Let $G$ be a group and $x \in G$ an element of finite order, say $|x|…
Every finite group of even order contains an element of order 2
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.31 Let $G$ be a finite group of even order. Prove that $G$ contains an element of…
Compute the order of an element in a direct product of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.30 Let $A$ and $B$ be groups and let $a \in A$ and $b \in B$ have…