Find a subgroup of order $6$ in $U(700).$
Attempt:
$U(700)=U(2^2.5^2.7)\thickapprox U(2^2) \oplus U(5^2)\oplus U(7)$
$\thickapprox \mathbb Z_2 \oplus \mathbb Z_{5^2-5} \oplus \mathbb Z_{7-7^0}$
$\thickapprox \mathbb Z_2 \oplus \mathbb Z_{20}\oplus\mathbb Z_6$
Hence, $U(700)\thickapprox \mathbb Z_2 \oplus \mathbb Z_{20} \oplus\mathbb Z_6$
Hence, if I have an isomorphism $\phi : \mathbb Z_2 \oplus \mathbb Z_{20} \oplus\mathbb Z_6 \mapsto U(700)$ such that If $(x,y,z ) \in \mathbb Z_2 \oplus \mathbb Z_{20} \oplus\mathbb Z_6$
, then : $ \phi (x,y,z ) = a \in U(700)$ , then $a$ has order $=6$.
Then, $\langle a \rangle$ forms a subgroup of order 6 in $U(700)$ .
By the property of isomorphism, this means the element $(x,y,z ) \in \mathbb Z_2 \oplus \mathbb Z_{20} \oplus\mathbb Z_6$ also must have order $6$.
One such element is $(0,0 ,1)$ . Now, if i know the isomorphism $\phi$, I can calculate the value of $\phi(0,0 ,1)$ .
But, how do I know what is this isomorphism? Thank you, help will be appreciated.
What about taking $H = \left\{ x \in U(700) | x \equiv 1 \mod 100 \right\}$