Exercises

1. Prove that there is no simple group of order $204$.

2. Prove that there is no simple group of order $18$.

3. Let $G$ be a finite group such that $\vert G\vert = pq$, where $p$ and $q$ are distinct primes such that $p\not\vert (q-1)$ and $q\not\vert (p-1)$. Then prove that $G$ is cyclic.

   small (HINT: Mimic the proof given in the text that any group of order $15$ is cyclic.)

4. Find all the $3$- and $2$-Sylow subgroup's for $A_4$.

   (HINT: The table for $A_4$ given in Section 6.1 may be helpful. Also recall something about $V_4$ in $A_4$.)

5. In the text, it was shown that if $G$ is a finite abelian group of order $n$, then for each $d > 0$ such that $d\vert n$, $G$ has a subgroup of order $d$. Does this imply that $G$ has an element of order $d$? WHY or WHY NOT?

   (HINT: $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$.)