Generating sets for subgroups of dihedral group:D8
This article gives specific information, namely, generating set, about a particular group, namely: dihedral group:D8.
View generating set of particular groups | View other specific information about 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 (not necessarily distinct) elements picked uniformly at random and independently of each other from a finite group
, the probability that they all live in a fixed subgroup of index
is
.
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 (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.
Automorphism class of subgroups | List of all subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Probability that an element is in a fixed subgroup of this automorphism class(= reciprocal of index) | Probability that an element generates a fixed subgroup of this automorphism class (obtained by Mobius inversion on preceding column) | Size of automorphism class | Probability that the element generates a subgroup in this automorphism class |
---|---|---|---|---|---|---|---|---|
trivial subgroup | ![]() |
trivial group | 1 | 8 | 1/8 | 1/8 | 1 | 1/8 |
center | ![]() |
cyclic group:Z2 | 2 | 4 | 1/4 | 1/8 | 1 | 1/8 |
other subgroups of order two | ![]() ![]() |
cyclic group:Z2 | 2 | 4 | 1/4 | 1/8 | 4 | 1/2 |
Klein four-subgroups | ![]() ![]() |
Klein four-group | 4 | 2 | 1/2 | 0 | 2 | 0 |
cyclic maximal subgroup | ![]() |
cyclic group:Z4 | 4 | 2 | 1/2 | 1/4 | 1 | 1/4 |
whole group | ![]() |
dihedral group:D8 | 8 | 1 | 1 | 0 | 1 | 0 |
Total | -- | -- | -- | -- | -- | -- | -- | 1 |
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.
Automorphism class of subgroups | List of all subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Probability that the elements are in a fixed subgroup of this automorphism class(= reciprocal square of index) | Probability that the elements generates a fixed subgroup of this automorphism class (obtained by Mobius inversion on preceding column) | Size of automorphism class | Probability that the elements generates a subgroup in this automorphism class |
---|---|---|---|---|---|---|---|---|
trivial subgroup | ![]() |
trivial group | 1 | 8 | 1/64 | 1/64 | 1 | 1/64 |
center | ![]() |
cyclic group:Z2 | 2 | 4 | 1/16 | 3/64 | 1 | 3/64 |
other subgroups of order two | ![]() ![]() |
cyclic group:Z2 | 2 | 4 | 1/16 | 3/64 | 4 | 3/16 |
Klein four-subgroups | ![]() ![]() |
Klein four-group | 4 | 2 | 1/4 | 3/32 | 2 | 3/16 |
cyclic maximal subgroup | ![]() |
cyclic group:Z4 | 4 | 2 | 1/4 | 3/16 | 1 | 3/16 |
whole group | ![]() |
dihedral group:D8 | 8 | 1 | 1 | 3/8 | 1 | 3/8 |
Total | -- | -- | -- | -- | -- | -- | -- | 1 |