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Let $A$ be a set of all complex numbers $z$ such that $z^24=1$ and let $B$ be the set of all complex numbers $w$ such that $w^54=1.$ That is: \begin{align*}A&=\{z\;|\;z^{24}=1\}\\ B&=\{z\;|\;z^{54}=1\}\\ \end{align*} Finally, let $C$ be the set of all complex numbers that can be formed by multiplying an element of $A$ by an element of $B$:

$C=\{zw\;|\;z\in A,w\in B\}.$

How many distinct elements are there in $C$?

I'm pretty stuck on this problem and don't know where to go. Any help would be very much appreciated!

majorbill
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1 Answers1

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This one gets easier after looking at it in its abstract content.

You have a finite abelian group $G$ (the one Jack Yoon mentioned in his comments) and two subgroups $A,B \subset G$. The question is: How to determine |AB|?

The answer is the exact sequence

$$1 \to A \cap B \to A \times B \to AB \to 1$$

with the first map given by $x \mapsto (x,x^{-1})$ (the second map should be more obvious).

This shows $|AB| = \frac{|A||B|}{|A \cap B|}$. Note that $|A \cap B|$ is often easier to determine than $|AB|$, especially in your case.

MooS
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