Let G be a group. Let a,b,c denote elements of G, and let e be the nov element of G. 1. Prove that if ab=e, then ba=c (Hint: See theorem 2. 2. Prove that if abc=e, then cab=e and bca = e. 3. State a generalization of pants 1 and 2.

Question
Exponential growth and decay
asked 2020-12-14
Let G be a group. Let a,b,c denote elements of G, and let e be the nov element of G.
1. Prove that if ab=e, then ba=c (Hint: See theorem 2.
2. Prove that if abc=e, then cab=e and bca = e.
3. State a generalization of pants 1 and 2.

Answers (1)

2020-12-15
Let ab=e. Multiplying from the left by \(\displaystyle{a}^{{-{{1}}}}\) yields \(\displaystyle{a}^{{-{{1}}}}{a}{b}={a}^{{-{{1}}}}{e}\to{e}{b}={a}^{{-{{1}}}}\to{b}={a}^{{-{{1}}}}\)
Therefore, \(\displaystyle{b}{a}={a}^{{-{{1}}}}{a}={e},\) as required.
2,We can use 1.: \(\displaystyle{a}{b}{c}={e}\to{\left({a}{b}\right)}{c}={e}\to{c}{\left({a}{b}\right)}={e}\to{c}{a}{b}={e}\) and \(\displaystyle{a}{b}{c}={e}\to{a}{\left({b}{c}\right)}={e}\to{\left({b}{c}\right)}{a}={e}\to{b}{c}{a}={e}\)
3.If x1,x2...xn are such that \(\displaystyle{x}{1}{x}{2}\ldots{x}{n}={e}\), then \(\displaystyle{x}{k}{x}{k}+{1}\ldots{x}{n}{x}{1}\ldots{x}{k}-{1}={e}\)
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Relevant Questions

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The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
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