Answered: Prove that G = <x, y | x2 = yn= e, xyx…

Name: Prove that G = is isomorphic to Dn.
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Description: Prove that G = is isomorphic to Dn.

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Prove that G = <x, y | x2 = yn= e, xyx = y-1> is isomorphic to Dn.

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter6: Exponential And Logarithmic Functions

Section6.7: Exponential And Logarithmic Models

Problem 27SE: Prove that bx=exln(b) for positive b1 .

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

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Question

Prove that G = <x, y | x² = yⁿ= e, xyx = y^-1> is isomorphic to D_n.

Expert Solution

Step 1

We have to prove that $G = 〈x, y | x^{2} = y^{n} = e, x y x = y^{- 1}〉$ is isomorphic to $D_{n}$ .

That is We have to prove that $G ≅ D_{n}$ .

First of all we will prove that $〈x, y〉 = 〈x, x y〉$ .

Clearly $x$ and $x y$ belongs to $〈x, y〉$ .

So, $〈x, x y〉 \subseteq 〈x, y〉$

Also $x$ and $x^{- 1} (x y) = y$ belongs to $〈x, x y〉$

Hence, $〈x, y〉 = 〈x, x y〉$ .

Also,

$\begin{matrix} {(x y)}^{2} = (x y) (x y) \\ = (x y x) y (Since, x y x = y^{- 1}) \\ = y^{- 1} y \\ = e \end{matrix}$

Hence, ${(x y)}^{2} = e$ .

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