Generating sets for subgroups of dihedral group:D8

This article provides basic information on various choices of generating sets for subgroups of dihedral group:D8. It builds on basic information available at element structure of dihedral group:D8 and subgroup structure of dihedral group:D8.



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.