## Every quotient of an abelian group is abelian

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.3 Let $A$ be an abelian group and let $B \leq A$. Prove that $A/B$ is abelian. Give an example of...

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# eMathPage

# Tagged: Quaternion Group

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Every quotient of an abelian group is abelian

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Compute the centralizers of each element in Sym(3), Dih(8), and the quaternion group

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Identify quaternion group as a subgroup of the general linear group of dimension 2 over complex numbers

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Dih(8) and the quaternion group are not isomorphic

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Find a presentation for the quaternion group

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Write out the Cayley table for Sym(3), Dih(8), and the quaternion group

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Compute the order of each element in the quaternion group

Free solutions to math textbooks

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.3 Let $A$ be an abelian group and let $B \leq A$. Prove that $A/B$ is abelian. Give an example of...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.4 For each of the groups $S_3$, $D_8$, and $Q_8$, compute the centralizer of each element and find the center of...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.26 Let $i$ and $j$ be the generators of $Q_8 = \langle i, j \ |\ i^4 = j^4 = 1,...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.7 Prove that $D_8$ and $Q_8$ are not isomorphic. Solution: We saw in Exercise 1.5.2 that $D_8$ has five elements of...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.3 Find a set of generators and relations for $Q_8$. Solution: Let $$X = \langle x, y \ |\ x^4 =...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.2 Write out the group tables for $S_3$, $D_8$, and $Q_8$. Solution: Group table for $S_3$: 1 (1 2) (1...

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.1 Compute the order of each of the elements in $Q_8$. Solution: $x$ Reasoning Order 1 1 is the identity. 1...