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The order of the group $G$, meet the following conditions: $1 where n is a natural number.

For each 2 sub groups $H_1$, $H_2$ of $G$, if $H_1 \neq H_2$ then $\gcd(|H_1|,|H_2|)=1$. (gcd = greatest common divisor)

Prove that the order of $G$ is a prime number and the group is cycle.

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    What do you mean by a neutral number? Since this looks like a homework problem, what have you tried?2012-07-16
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    The order of G is not infinite.2012-07-16
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    Note that the condition you mention also has to hold when $H_1 = G$ and $H_2$ is any proper subgroup of $G$. What does Lagrange then tell you?2012-07-16
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    @Lag: Those are called **natural** numbers, not "neutral" numbers.2012-07-16

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