Generating sets for subgroups of alternating group:A4

Definition
This article provides summary information on various choices of generating sets for subgroups of alternating group:A4. It builds on basic information available at element structure of alternating group:A4 and subgroup structure of alternating group:A4.



Probability of generation
The rule is as follows. Given $$k$$ (not necessarily distinct) elements picked uniformly at random and independently of each other from a finite group $$G$$, the probability that they all live in a fixed subgroup of index $$d$$ is $$1/d^k$$.

Using this and a form of Mobius inversion on the subgroup lattice, it is possible to compute the probability that they generate a fixed subgroup of index $$d$$ (we basically need to subtract off probabilities for smaller subgroups). This probability can also be computed as a product:

(Probability that the elements lie in the subgroup) times (Probability of generating the subgroups conditional to lying in the subgroup)

The former probability depends only on the index of the subgroup and the latter probability depends only on the isomorphism class of the subgroup.