Answered: In the frieze group F7, write x-3zxyz…

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In the frieze group F7, write x-3zxyz in the form xnymz k.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 32E: If a is an element of order m in a group G and ak=e, prove that m divides k.

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Question

In the frieze group F₇, write x^-3zxyz in the form xⁿy^mz ^k.

Expert Solution

Step 1

It is known that $F_{7} = \{x^{n} y^{m} z^{k} ∥ n \in Z, m = 0, k = 0 o r k = 1\}$ and $x$ is the translation of $R^{2}$ , and it map $(u, v)$ to $(u + a, v)$ for some constant $a$ .

Also, $y$ is the horizontal reflection of $R^{2}$ , and it map $(u, v)$ to $(u, - v)$ .

Also, $z$ is the vertical reflection of $R^{2}$ , and it map $(u, v)$ to $(- u, v)$ .

So, it can be calculated that both $y z$ and $z y$ map $(u, v)$ to $(- u, - v)$ .

So, it results,

$y z = z y \dots (1)$

Also it can be concluded that both $x y$ and $y x$ map $(u, v)$ to $(u + a, v)$ .

so, it results,

$x y = y x \dots (2)$

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