Answered: Let a be a group element and lal = o0.… | bartleby

Advanced Engineering Mathematics

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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Question

Let a be a group element and lal = o0. Complete the following
statement: la'l = la'l if and only if ....

Expert Solution

Step 1

Given that $a$ be a group element and $|a| = \infty$ .

We need to complete the statement: $|a^{i}| = |a^{j}|$ if and only if ......

Step 2

If part:

$|a^{i}| = |a^{j}|$

Let $|a^{i}| = |a^{j}| = k$

We get,

$\begin{array}{rcl} a^{i} \cdot a^{i} \cdot . . . . . \cdot a^{i} (k - t i m e s) & = & a^{j} \cdot a^{j} \cdot . . . . \cdot a^{j} (k - t i m e s) = e (i d e n t i t y) \\ {(a^{i})}^{k} & = & {(a^{j})}^{k} \\ a^{i k} & = & a^{j k} \\ a^{i k} \cdot a^{- j k} & = & a^{j k} \cdot a^{- j k} (a^{j k} \in G \Rightarrow a^{- j k} \in G) \\ a^{i k - j k} & = & a^{0} \\ a^{i k - j k} & = & e \end{array}$

Since $|a| = \infty$ , we get $i k - j k = 0$

$i k - j k = 0 i k = j k i = j$

If $|a^{i}| = |a^{j}|$ and $|a| = \infty$ , then $i = j$ .

Conversely:

If $i = j$

$a^{i} = a^{j}$ implies $|a^{i}| = |a^{j}|$

If $i = j$ then $|a^{i}| = |a^{j}|$ .

Therefore, $|a| = \infty$ and $|a^{i}| = |a^{j}|$ if and only if $i = j$ .

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