## 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: Order

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

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Perform explicit computation in a quotient of a quasi-dihedral group

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Compute the order of a quotient group element

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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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If a prime power power of a group element is trivial, then the order of the element is a prime power

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The order of a product of commuting group elements divides the least common multiple of the orders of the elements

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

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If the group and an element have the same finite order then the group is cyclic

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The order of a general linear group over a finite field is bounded above

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 3.1 Exercise 3.1.18 Let $G = QD_{16}$ be the quasidihedral group presented by $$\langle \sigma, \tau \ |\ \sigma^8 = \tau^2 = 1,...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.5 Let $G$ be a group and $N$ a normal subgroup of $G$. Prove that the order of the element $gN$...

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 2.3.20 Let $p$ be a prime and $n$ a positive integer. Show that if $x$ is an element of a group...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.16 Let $G$ be a group with $x,y \in G$. Assume $|x| = n$ and $|y| = m$. Suppose that $x$...

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 2.3.2 Let $G$ be a finite group and let $x \in G$. Prove that if $|x| = |G|$ then $G =...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.6 Let $F$ be a field. If $|F| = q$ is finite show that $|GL_n(F)| < q^{n^2}$. Solution: Clearly $GL_n(F)$ is...