Probability question in group-theoretic context

Question

I'd like to know the probability that four randomly chosen symmetries of the cube generate the whole octahedral group $C_{2}\times S_{4}$... is there some quick way of working this out, i.e. avoiding brute combinatorial listings?

Answer 1

Here's one approach. A subset of a group generates the group if and only if it isn't contained in any proper subgroup. If the group is finite, this is equivalent to saying that the subset isn't contained in any maximal subgroup. The maximal subgroups of the octahedral group are as follows:

1. $S_4$ (just rotations; order 24)

2. $C_2 \times A_4$ (order 24)

3. $(\{0\} \times A_4) \cup (\{1\} \times (S_4 \setminus A_4))$ (order 24)

4-6. $C_2 \times D_8$ (stabilizers of a pair of opposite faces; order 16, three conjugates)

7-10. $C_2 \times S_3$ (stabilizers of a pair of opposite vertices; order 12, four conjugates)

For any subgroup $H \leq G$, the probability that all four elements are contained in $H$ is just $\left(\frac{|H|}{|G|}\right)^4$. So if you work out the sizes of all possible intersections between the maximal subgroups listed above, then you can get an answer by the inclusion-exclusion principle. You could (for example) do this in GAP, which also has nice commands for finding maximal subgroups.