# Tagged: Group

## The set of all group automorphisms of a fixed group is a group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.20 Let $G$ be a group and let $\mathsf{Aut}(G)$ be the set of all isomorphisms $G \rightarrow G$. Prove that $\mathsf{Aut}(G)$...

## 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 order 2. Solution: Let...

## 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 finite order. Prove that...

## 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 multiplication; i.e., (a_1, b_1)...

## 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 $n = st$ for...