Statements (102)
| Predicate | Object |
|---|---|
| gptkbp:instance_of |
gptkb:musical_group
|
| gptkbp:action_on_set |
{1, 2, 3}
|
| gptkbp:character_traits |
D_3
|
| gptkbp:difficulty |
gptkb:true
|
| gptkbp:element |
(1) (12) (13) (23) (123) (132)
|
| gptkbp:has |
2 cycles
6 elements 1 identity element 2 elements of order 3 3 elements of order 2 3 permutations of order 2 3 transpositions a dihedral structure a non-trivial automorphism group a subgroup of order 2 one subgroup of order 3 two generators |
| gptkbp:has_a_cayley_graph |
with 6 vertices
|
| gptkbp:has_a_center |
which is trivial
|
| gptkbp:has_a_normal_subgroup |
A_3
|
| gptkbp:has_conjugacy_classes |
gptkb:3
|
| gptkbp:has_graph_representation |
gptkb:Cayley_graph
|
| gptkbp:has_group_order |
gptkb:6
|
| gptkbp:has_group_presentation |
⟨a, b | a^3 = b^2 = 1, bab = a^2⟩
|
| gptkbp:has_identity_element |
(1)
|
| gptkbp:has_inverse_element |
gptkb:true
|
| gptkbp:has_permutation_group |
gptkb:true
|
| gptkbp:has_produced |
two elements
(12) (123) two transpositions |
| gptkbp:has_representations |
2-dimensional representations
|
| gptkbp:has_subgroup |
gptkb:1
gptkb:3 gptkb:6 gptkb:S_3 gptkb:S_4 2 A_3 (1) (12) (1) (123) (1) (132) S_n for n ≥ 3 |
| gptkbp:has_transpositions |
gptkb:3
|
| gptkbp:has3_cycles |
2
|
| https://www.w3.org/2000/01/rdf-schema#label |
S 3
|
| gptkbp:is |
false
abelian smallest non-abelian group |
| gptkbp:is_a_center_for |
(1)
|
| gptkbp:is_a_finite_group |
of order 6
|
| gptkbp:is_a_non-abelian_group |
with 3 conjugacy classes
|
| gptkbp:is_a_permutation_group |
of degree 3
|
| gptkbp:is_a_simple_group |
in terms of its structure
|
| gptkbp:is_a_symmetric_group |
on 3 elements
|
| gptkbp:is_abelian |
false
|
| gptkbp:is_associative |
gptkb:true
|
| gptkbp:is_closed_under_operation |
gptkb:true
|
| gptkbp:is_related_to |
the concept of symmetry
|
| gptkbp:is_represented_in |
symmetric functions
symmetric matrices 2x2 permutation matrices permutations of three objects the symmetric group on three letters |
| gptkbp:is_the_smallest |
non-abelian group
|
| gptkbp:is_used_in |
gptkb:crypt
gptkb:topology gptkb:computer_science gptkb:Mathematics gptkb:musical_group physics geometry number theory representation theory abstract algebra category theory mathematical logic combinatorial problems combinatorics the study of group actions |
| gptkbp:isomorphic_to |
D_3
|
| gptkbp:members |
has applications in physics
is used in cryptography can be generated by two transpositions can be represented by permutations can be visualized with a triangle has a geometric interpretation has a non-abelian structure has a trivial center is a basic example in the classification of groups is a finite symmetric group is a key example in abstract algebra is a model for symmetry operations is fundamental in group theory is often the first example of a non-abelian group is studied in algebra is used in combinatorial designs order 6 with non-commutative multiplication |
| gptkbp:order |
gptkb:3
gptkb:6 2 |
| gptkbp:permutation_representation |
3! = 6
|