Order Equal to 300

Let $\mid G \mid = 300 = 2^{2} \times 3 \times 5^{2}$. Sylow's Theorem assures the following:

$$\begin{tabular}{\vert c\vert c\vert l\vert} \hline prime & Sy... ...& 1, 4, 10, 25 \\ \hline 5 & 25 & 1, 6 \\ \hline \end{tabular}$$

Lets look at n₅. The Index Theorem will not allow n₅ = 6because $\mid G \mid$ does not dived 6!. Therefore, n₅ = 1 and so a Syl₅ subgroup is normal in G. Thus, G is not simple!