Let $G, G'$ are two groups, and $f$ is a group homomorphism from $G$ to $G'$. is the following inclusion satisfied? $f^{-1}(x).f^{-1}(y)\subset f^{-1}(x.y)$ for all $x,y \in G'$ and why? thanks.
about a property of group homomorphism
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$\begingroup$
group-theory
finite-groups
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0This is all about unraveling definitions. If something is in $f^{-1}(x)$, say $a$, what does that mean? What is a homomorphism? – 2017-01-12
1 Answers
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Given $a\in f^{-1} (x)$ and $b\in f^{-1}(y) $ we have $f (ab)=f (a)f (b)=xy $ and so $ab\in f^{-1}(xy) $.
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0Ok. And what does it lead if $f^{-1}(xy)$ is empty? – 2017-01-12
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0This argument shows that if $f^{-1}(x) $ and $f^{-1}(y) $ aren't both empty then $f^{-1}(xy) $ is also not – 2017-01-12
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0Actually I have $f^{-1}(xy)$ is empty, can I conclude that one of $f^{-1}(x), f^{-1}(y)$ is empty ? – 2017-01-12
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0Yes! It is equivalent to what I did. – 2017-01-12