If a is an element of order m in a group G and a^(k) = e, prove that m divides k.
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Do a long division on the exponent.
Originally Posted by FancyMouse Do a long division on the exponent. I don't understand what you mean. Can you explain?
Originally Posted by rainyice I don't understand what you mean. Can you explain? Let k=rm+q where r is an integer and 0<=q<m. What can you say about $\displaystyle a^q$?
Originally Posted by FancyMouse Let k=rm+q where r is an integer and 0<=q<m. What can you say about $\displaystyle a^q$? it has infinite order?
$\displaystyle k=rm+q $, where $\displaystyle q<m $. So, $\displaystyle 1=a^k=a^{rm+q}=(a^{m})^r\cdot a^q = a^q $ Since $\displaystyle q<m $, what is the only such $\displaystyle q $ that gives $\displaystyle a^q = 1 $?
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