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(A) Show that $(\Bbb Z/5\Bbb Z)^\times$ is isomorphic to $(\Bbb Z/10\Bbb Z)^\times$

(B) $(\Bbb Z/8\Bbb Z)^\times$ is not isomorphic to $(\Bbb Z/10\Bbb Z)^\times$.

For (A) since these groups are cyclic hence isomorphic exist between these groups

Am I right?

I don't have idea about (B)

  • 1
    What is $x$?${}{}$ Are you using $x$ to designate $\times$? I mean, are you trying to ask, e.g. whether $(\mathbb Z/5)^\times \cong (\mathbb Z/10)^\times$? – 2017-02-09
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    yes that is right – 2017-02-09

1 Answers 1

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In $(\Bbb Z/8\Bbb Z)^{\times}=\{1,3,5,7\}$ any element $x$ verify $x^2=1$. But in $(\Bbb Z/10\Bbb Z)^{\times}$, $3^2=9\neq 1$.