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Find a subgroup of order $6$ in $U(700).$

My attempt:

$U(700)=U(2^2.5^2.7)\simeq U(2^2)\oplus U(5^2)\oplus U(7)\simeq\mathbb Z_2\oplus\mathbb Z_{5^2-5}\oplus\mathbb Z_{7-7^0}\simeq\mathbb Z_2\oplus\mathbb Z_{20}\oplus\mathbb Z_6\\\mathbb Z_2\le\mathbb Z_2\\(0)\le\mathbb Z_{20}\\(2)\le\mathbb Z_6\\\implies\mathbb Z_2\oplus(0)\oplus(2)\le U(700)~and~|\mathbb Z_2\oplus(0)\oplus(2)|=6$

Is my attempt correct?

Community
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Sriti Mallick
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    Sort-of; you might have to specify what subgroup of $U(700)$ that actually corresponds to, rather than working in an isomorphic group. (Also, why not just choose $0\oplus0\oplus Z_6$?) – anon Jun 21 '13 at 02:24

1 Answers1

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Hint: Try to find an integer $n$ such that $n\equiv 1\pmod{100}$, and $n\equiv a\pmod 7$ such that $a$ is a generator of $U(7)$.

Jyrki Lahtonen
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