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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…

## 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…