Answered: Let x and y be two vertices of a Cayley…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let x and y be two vertices of a Cayley digraph. Explain why two
paths from x to y in the digraph yield a group relation—that is, an equation of the form a₁a₂ . . . a_m = b₁b₂ . . . b_n, where the a_i’s and
b_j’s are generators of the Cayley digraph.

Expert Solution

Step 1

It is given that $x$ and $y$ be two vertices of a Cayley digraph.

We can write two paths as $(a_{1}, a_{2}, a_{3}, \dots a_{n})$ and $(b_{1}, b_{2}, b_{3}, \dots b_{n})$ , where $a_{i}, b_{j}$ are both generators.

So, both the paths start from $x$ .

It ca be observed that end points of the two paths should be $x (a_{1}, a_{2}, a_{3}, \dots a_{n})$ and $x (b_{1}, b_{2}, b_{3}, \dots b_{n})$ respectively.

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