A finite direct product is abelian if and only if each direct factor is abelian
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.29 Let $A$ and $B$ be groups. Prove that $A \times B$ is abelian if and only…
A finite direct product of groups is a group under componentwise multiplication
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.28 Let $A$ and $B$ be groups. Verify that $A \times B$ is a group under componentwise…
In a group, the set of powers of a fixed element is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.27 Let $G$ be a group and let $x \in G$. Prove that $A = \{ x^n…
Subsets of a group which are closed under multiplication and inversion are groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.26 Let $(G, \star)$ be a group. Show that a nonempty subset $H \subseteq G$ which is…
A group that every element has order 2 is abelian
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.25 Let $G$ be a group. Prove that if $x^2 = 1$ for all $x \in G$,…
Powers distribute over products of commuting group elements
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.24 Let $G$ be a group and let $a,b \in G$ such that $ab = ba$. Prove…
Characterization of the order of powers of a group element
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.23 Let $G$ be a group. Suppose $x \in G$ with $|x| = n < \infty$. If…
Conjugation preserves order
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.22 Let $G$ be a group and let $x,g \in G$. Prove that $|x| = |g^{-1} x…
Every group element of odd order is an odd power of its square
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.21 Let $G$ be a finite group and let $x \in G$ be an element of order…
A group element and its inverse have the same order
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.20 Let $G$ be a group and let $x \in G$. Prove that $x$ and $x^{-1}$ have…