This is Daily Work 16.

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Let \(G\) be a mysterious finite group of order 144, and suppose that \(H\) is a subgroup of \(G\).

Which of the following cannot be the order of \(H\)?
The symmetric group \(S_4\) has order 24. 

Does it have a subgroup of order 12?
Which line of this proof contains an error?

Let \(G\) be a group of order 9. We'll show \(G\) is cyclic.
I. Let \(a\in G\) be an arbitrary element.
II. By Lagrange's theorem, \( |a| = 9 \).
III. Thus \( \langle a \rangle \) is a subgroup of order 9.
IV. So \( \langle a \rangle = G \) and \(G\) is cyclic.

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