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I would like to understand Sylow $p$-subgroups.

I understand Sylow discovered the converse of Lagrange's Theorem is true for a prime divisor of finite group.

I am trying to understand following example.

$\textbf{Example:}$ Let $G=\Bbb {Z_{12}}$ be group with additive operation.

So $|G|=12=2^23$. Then $G$ has Sylow $2$-subgroup(s) and Sylow $3$-subgroup(s). Ok I know $n_2|3$ and $n_2\equiv 1$ (mod $2$), where $n_2$ the numbers of Sylow $2$-subgroup. So, $n_2=1$ or $n_2=3$.

I can't maintain solution. I have similar question for Sylow $3$-subgroup(s).

In general, I want to know following question answer:

• Is there a general solution algorithm?

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