Original question and givens
Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=3. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).
G001 — Use complete finite group Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]], identity=0. Certify all group axioms and inverses before the complete all-element commutation and conjugation audits. Candidate element=3. Give (element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary).
Step-by-step derivation
S001 · implies
Before
source roles = [center-abelian-conjugacy-boundary]
After
derived = [[element, central, conjugacy_class, singleton_class, central_singleton_equivalence, center, abelian, center_equals_group, abelian_center_equivalence, noncommuting_element_or_null, boundary]]
Conditions: none
List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding.
Step identifiers and dependencies
Rule: source_ordered_fields_without_JS · Uses: G001
S002 · implies
Before
source roles = [Center ORIGINAL group, [[0, 1, 2, 3, 4, 5], [1, 2, 0, 4, 5, 3], [2, 0, 1, 5, 3, 4], [3, 5, 4, 0, 2, 1], [4, 3, 5, 1, 0, 2], [5, 4, 3, 2, 1, 0]], 0]
After
derived = [6, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5]], Yes]
Conditions: none
Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0.
Step identifiers and dependencies
Rule: Group_complete_table_actual_identity_bound64 · Uses: G001, S001
S003 · implies
Before
source roles = [Center ORIGINAL group, 6, 0]
After
derived = [[[Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes]], 216, [0, 2, 1, 3, 4, 5]]
Conditions: none
Each pair-matrix entry checks every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides.
Step identifiers and dependencies
Rule: Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · Uses: G001, S002
S004 · implies
Before
source roles = [0, [0, 1, 2, 3, 4, 5]]
After
derived = [[0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [Yes, Yes, Yes, Yes, Yes, Yes], Yes]
Conditions: none
For each z compare zg and gz across the full row,not the centralizer of one chosen g.
Step identifiers and dependencies
Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S003
S005 · implies
Before
source roles = [1, [0, 1, 2, 3, 4, 5]]
After
derived = [[1, 2, 0, 4, 5, 3], [1, 2, 0, 5, 3, 4], [Yes, Yes, Yes, No, No, No], No]
Conditions: none
For each z compare zg and gz across the full row,not the centralizer of one chosen g.
Step identifiers and dependencies
Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S004
S006 · implies
Before
source roles = [2, [0, 1, 2, 3, 4, 5]]
After
derived = [[2, 0, 1, 5, 3, 4], [2, 0, 1, 4, 5, 3], [Yes, Yes, Yes, No, No, No], No]
Conditions: none
For each z compare zg and gz across the full row,not the centralizer of one chosen g.
Step identifiers and dependencies
Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S005
S007 · implies
Before
source roles = [3, [0, 1, 2, 3, 4, 5]]
After
derived = [[3, 5, 4, 0, 2, 1], [3, 4, 5, 0, 1, 2], [Yes, No, No, Yes, No, No], No]
Conditions: none
For each z compare zg and gz across the full row,not the centralizer of one chosen g.
Step identifiers and dependencies
Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S006
S008 · implies
Before
source roles = [4, [0, 1, 2, 3, 4, 5]]
After
derived = [[4, 3, 5, 1, 0, 2], [4, 5, 3, 2, 0, 1], [Yes, No, No, No, Yes, No], No]
Conditions: none
For each z compare zg and gz across the full row,not the centralizer of one chosen g.
Step identifiers and dependencies
Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S007
S009 · implies
Before
source roles = [5, [0, 1, 2, 3, 4, 5]]
After
derived = [[5, 4, 3, 2, 1, 0], [5, 3, 4, 1, 2, 0], [Yes, No, No, No, No, Yes], No]
Conditions: none
For each z compare zg and gz across the full row,not the centralizer of one chosen g.
Step identifiers and dependencies
Rule: Center_EVERY_z_commutes_against_EVERY_g · Uses: G001, S008
S010 · implies
Before
source roles = [Center ACTUAL Z, [0], 0]
After
derived = [Yes, [[0, 0, 0, Yes]], Yes, [[0, 0, Yes]], Yes, Yes, [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes]], Yes]
Conditions: none
Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name.
Step identifiers and dependencies
Rule: Subgroup_actual_identity_product_inverse_ALL_conjugates · Uses: G001, S009
S011 · implies
Before
source roles = [[0], [[Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, No, No, No], [Yes, Yes, Yes, No, No, No], [Yes, No, No, Yes, No, No], [Yes, No, No, No, Yes, No], [Yes, No, No, No, No, Yes]]]
After
derived = [Yes, Yes, central z: g*z*g^-1=z for EVERY g]
Conditions: none
Identity commutes with every g;products and inverses of all-g commuting elements do too.Right multiplying gz=zg gives gzg^-1=z,so central elements are fixed pointwise and the center is normal.No characteristic or class-equation claim.
Step identifiers and dependencies
Rule: Center_GENERAL_closure_and_pointwise_fixed_conjugation · Uses: G001, S010
S012 · implies
Before
source roles = [3, [[0, 3], [1, 5], [2, 4], [3, 3], [4, 5], [5, 4]], [0]]
After
derived = [No, [3, 4, 5], No, Yes, No, No, Yes, 1]
Conditions: none
Because gxg^-1=x iff gx=xg,centrality is equivalent to a singleton conjugacy class.Z=G iff all elements commute.BothFalse still give True equivalence.Noncentral elements retain the first original-index noncommuter;central ones give null,not0/False.
Step identifiers and dependencies
Rule: Center_actual_singleton_iff_and_abelian_iff_FLAGS_FIRST_noncommuter · Uses: G001, S011