Short exact sequence with binary tetrahedral group does not split The following is a short exact sequence, where $T$ is the binary tetrahedral group (equivalently the Hurwitz units), and $Q$ is the quotient of $T$ by $\mathbb{Z}/2$. $1 \rightarrow \mathbb{Z}/2 \rightarrow T \rightarrow Q \rightarrow 1$. Wikipedia tells me that this doesn't split-- i.e. that there is no map $\phi: Q \rightarrow T$ which is the identity on $Q$ when composed with the map from $Q$ to $T$. Why is this true, preferably from as elementary a perspective as possible? Thanks!

If the extension split, then the image of $Q$ ($\cong A_4$) inside $T$ under a splitting homomorphism would be normal, as its index would be equal to $2$. However, the only normal subgroups are the centre of order $2$ and the quaternion subgroup of index $3$.

One way to see that $T$ has no subgroup of index $2$ is to note that, if it did, then by the correspondence theorem, so too would $Q\cong A_4$. But, $A_4$ has no subgroup of index $2$ (i.e., of order $6$). This is often cited as a counter-example to the converse of Lagrange's Theorem.