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…
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 $x \in G$,…
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…