# This is Daily Work 16.

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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