Generating sets for subgroups of alternating group:A4

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This article gives specific information, namely, generating sets for subgroups, about a particular group, namely: alternating group:A4.
View generating sets for subgroups of particular groups | View other specific information about 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.

A4latticeofsubgroups.png

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.