2
$\begingroup$

Let $G$ be a group such that $|G| = 2012$, how would you classify, up to isomorphism, all groups $G$?

Clearly $2012 = 503 \times 2 \times 2$ and so $G \cong C_{503} \times C_2 \times C_2$ but how would you find the others?

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    I would begin by figuring out how many Sylow 503-subgroups there are :-) Or, if you haven't covered the theory of Sylow subgroups yet, I would show that any such group has a normal subgroup of order 503.2012-06-10
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    Don't forget $C_{503}\times C_4$...2012-06-10
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    Jyrki Lahtonen, how do you figure that out? We have that if $p^nq = |G|$ where $p$ doesn't divide $q$ and $n$ is as large as possible then the number of Sylow $p$ subgroups satisfy $1\mod p$ so there are $1\mod 503$ subgroups?2012-06-10
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    But doesn't $1 \mod 503$ and $2\mod 503$ both divide 4?2012-06-10
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    @morphism, the number of Sylow 503-subgroups, call it $n_{503}$ must satisfy $$n_{503}\equiv 1\pmod{503}$$ and $$n_{503}\mid 2012.$$ The first congruence shows that $503\nmid n_{503}$, so the second criterion simplifies to $n_{503}\mid 4$ (often the Sylow theorems are stated to say this, because we can always make this same deduction). So $n_{503}$ is either 1, 2 or 4. Only the choice $n_{503}=1$ satisfies the first congruence.2012-06-10

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