Question

Prove that there is no simple group of order 12. [Hint: Show that one of the Sylow subgroups must be normal.]

Short answer

It is proved that there is no simple group of order 12.

Step-by-step solution

Prove there is no simple group of order 12

Let G be a group of order 12.

Rewrite it as:

$\begin{array}{rcl}O\left(G\right)& =& 12\\ & =& 2^2\times 3\end{array}$

So, has a Sylow 2-subgroup of order 4 and a Sylow 3-subgroup of order 3.

Here, $O\left(G\right)=2^3\times 3=p^2\times q\left[p=2,q=3,\&pandqaredistinctprime\right]$.

Therefore, by corollary 9.31, there is no simple group of order $p^{2}q=12$.

Hence, there is no simple group of order 12.