Facts about double cosets
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.10 Let $G$ be a group and $H,K \leq G$ subgroups. For each $x \in G$, define…
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…
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 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…
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…
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