Generating sets for subgroups of symmetric group:S3

This article provides summary information on various choices of generating set for subgroups of symmetric group:S3. It builds on basic information available at element structure of symmetric group:S3 and subgroup structure of symmetric group:S3.



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).

Generated by one element
Here, a single element is picked uniformly at random from the group.

Generated by two independent possibly equal elements
Here, two elements are picked uniformly at random from the group, independent of each other. They could be equal.

Generated by $$k$$ elements picked independently and uniformly at random
Below are given the expressions for general $$k$$.

Small generating sets for subgroups
For symmetric group:S3, the following are equivalent for any subset:


 * It is a generating set of minimum size for the subgroup it generates.
 * It is a minimal generating set for the subgroup it generates.

We list below all the small generating sets:

All other subsets fail each of the four questions, so they are not listed.