Exercises

1. Suppose $N \lhd A_n$ for $n \geq 5$ and that $(i,j,k)\in N$ where $i \not= 2$ and $j = 1$, show as in the proof of Theorem 6.3.2 that this again implies that $N =A_n$.

2. Go through the proof of Theorem 6.3.2 and answer all the questions asked there.

3. Use the main result of this section, i.e., Theorem 6.3.2, to give another example, different from that in the text or in exercise 4 of Section 6.2, showing the converse of Lagrange's Theorem is false.

4. Let $G$ be an abelian simple group. Prove $\vert G\vert = p$, $p$ a prime. (It is not assumed initially that $G$ is finite.)

5. Let $G$ be a non-abelian group s.t. $\vert G\vert = p^3$. Prove $\vert Z(G)\vert = p$.

6. If $\vert G\vert = pq$, where $p$ and $q$ are not necessarily distinct primes, prove $\vert Z(G)\vert = 1$ or $=pq$.