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## Exhibit the automorphisms of Z/(48)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.8 Let $Z_{48} = \langle x \rangle$. For which integers a does the map $\varphi_a$ defined by…

## Compute the number of generators of Z/(49000)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.5 Find the number of generators for $\mathbb{Z}/(49000)$. Solution: The number of generators of $\mathbb{Z}/(n)$ is $\varphi(n)$,…

## Find all generators of Z/(202)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.4 Find all generators for $\mathbb{Z}/(202)$. Solution: The generators of $\mathbb{Z}/(202)$ are precisely those (equivalence classes represented…

## Find the generators of Z/(48)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.3 Find all the generators in $\mathbb{Z}/(48)$. Solution: The generators of $\mathbb{Z}/(48)$ are precisely those (equivalence classes…

## If a finite group has a generating set containing two elements of order 2, then it is a dihedral group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.24 Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order…