## Perform explicit computation in a quotient of the modular group of order 16

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.19 Let $G = M = \langle u,v \ |\ u^2 = v^8 = 1, vu = uv^5 \rangle$ and let...

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# Tagged: Residue Group

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Perform explicit computation in a quotient of the modular group of order 16

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The group of units in $\mathbb Z/(2^n)$ is not cyclic for n at least 3

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Compute the order of 5 in the integers mod a power of 2

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Use the Binomial Theorem to compute the order of an element in the integers mod a prime power

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Compute the order of a cyclic subgroup in Z/(54)

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Find all generators of Z/(202)

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Find the generators of Z/(48)

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Compute the subgroup lattice of Z/(45)

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If n is composite, then Z/(n) is not a field

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Compute multiplicative orders in Z/(36)

Free solutions to math textbooks

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.19 Let $G = M = \langle u,v \ |\ u^2 = v^8 = 1, vu = uv^5 \rangle$ and let...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.23 Show that $(\mathbb{Z}/(2^n))^\times$ is not cyclic for any $n \geq 3$. (Hint: find two distinct subgroups of order 2.) Solution:...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.22 Let $n$ be an integer with $n \geq 3$. Use the Binomial Theorem to show that $$(1+2^2)^{2^{n-2}} = 1 \pmod...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.21 Let $p$ be an odd prime and $n$ a positive integer. Use the Binomial Theorem to show that $(1+p)^{p^{n-1}} =...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.10 What is the order of $\overline{30}$ in $\mathbb{Z}/(54)$? Write out all of the elements and their orders in $\langle \overline{30}...

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 by) integers a such...

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 represented by) integers $k$,...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.1 Find all subgroups of $G = \mathbb{Z}/(45)$, giving a generator for each. Describe the containments among these subgroups. Solution: The...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.4 Show that if $n$ is not prime, then $\mathbb{Z}/(n)$ is not a field. Solution: If $n$ is not prime, then...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.14 Find the orders of the following elements of the multiplicative group $(\mathbb{Z}/(36))^\times$: $\overline{1}$, $\overline{-1}$, $\overline{5}$, $\overline{13}$, $\overline{-13}$, and $\overline{17}$. Solution:...