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Question 29248: Close Every Group Obligation on the Specified Finite Carrier

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0057-29248 · q-20261013-advance-20261022-r05-finite-group-0098

Original question and givens

The complete finite Cayley table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.

  • G001 — The complete finite Cayley table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-str22-group-full-group-audit, group-full-group-audit]

After

derived = [[n, table, closed, associative, identities, inverse_rows, group]]

Conditions: none

Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning.

Step identifiers and dependencies

Rule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001

S002 · implies

Before

source roles = [5, 0, [3, 4, 0, 1, 2]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S001

S003 · implies

Before

source roles = [5, 1, [4, 0, 1, 2, 3]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S002

S004 · implies

Before

source roles = [5, 2, [0, 1, 2, 3, 4]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S003

S005 · implies

Before

source roles = [5, 3, [1, 2, 3, 4, 0]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S004

S006 · implies

Before

source roles = [5, 4, [2, 3, 4, 0, 1]]

After

derived = [[Yes, Yes, Yes, Yes, Yes]]

Conditions: none

Check EVERY cell of this complete Cayley row lies in Fin(n),preserving row/column labels.Closure qualifies a finite binary operation,not the remaining group axioms.

Step identifiers and dependencies

Rule: Group_EVERY_original_row_closure · Uses: G001, S005

S007 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 0, 0]

After

derived = [[1, 2, 3, 4, 0], [1, 2, 3, 4, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S006

S008 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 0, 1]

After

derived = [[2, 3, 4, 0, 1], [2, 3, 4, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S007

S009 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 0, 2]

After

derived = [[3, 4, 0, 1, 2], [3, 4, 0, 1, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S008

S010 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 0, 3]

After

derived = [[4, 0, 1, 2, 3], [4, 0, 1, 2, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S009

S011 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 0, 4]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S010

S012 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 1, 0]

After

derived = [[2, 3, 4, 0, 1], [2, 3, 4, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S011

S013 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 1, 1]

After

derived = [[3, 4, 0, 1, 2], [3, 4, 0, 1, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S012

S014 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 1, 2]

After

derived = [[4, 0, 1, 2, 3], [4, 0, 1, 2, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S013

S015 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 1, 3]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S014

S016 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 1, 4]

After

derived = [[1, 2, 3, 4, 0], [1, 2, 3, 4, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S015

S017 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 0]

After

derived = [[3, 4, 0, 1, 2], [3, 4, 0, 1, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S016

S018 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 1]

After

derived = [[4, 0, 1, 2, 3], [4, 0, 1, 2, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S017

S019 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 2]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S018

S020 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 3]

After

derived = [[1, 2, 3, 4, 0], [1, 2, 3, 4, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S019

S021 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 4]

After

derived = [[2, 3, 4, 0, 1], [2, 3, 4, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S020

S022 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 3, 0]

After

derived = [[4, 0, 1, 2, 3], [4, 0, 1, 2, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S021

S023 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 3, 1]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S022

S024 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 3, 2]

After

derived = [[1, 2, 3, 4, 0], [1, 2, 3, 4, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S023

S025 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 3, 3]

After

derived = [[2, 3, 4, 0, 1], [2, 3, 4, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S024

S026 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 3, 4]

After

derived = [[3, 4, 0, 1, 2], [3, 4, 0, 1, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S025

S027 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 4, 0]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S026

S028 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 4, 1]

After

derived = [[1, 2, 3, 4, 0], [1, 2, 3, 4, 0], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S027

S029 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 4, 2]

After

derived = [[2, 3, 4, 0, 1], [2, 3, 4, 0, 1], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S028

S030 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 4, 3]

After

derived = [[3, 4, 0, 1, 2], [3, 4, 0, 1, 2], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S029

S031 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 4, 4]

After

derived = [[4, 0, 1, 2, 3], [4, 0, 1, 2, 3], []]

Conditions: none

For EACH original(a,b),list both parenthesizations(a*b)*c and a*(b*c) for EVERY c=0..n-1 and ALL failures.Include repeated elements;neither sample nor retain only the first counterexample.

Step identifiers and dependencies

Rule: Group_ALL_third_coordinates_for_EACH_ordered_first_pair · Uses: G001, S030

S032 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]]]

After

derived = [[], Yes]

Conditions: none

The(a,b) loops and ALL c cover n³ ordered triples.Retain ALL bad triples in nested increasing order;only an empty list certifies associativity of THIS complete finite table,not an unknown structure.

Step identifiers and dependencies

Rule: Group_ALL_bad_triples_nested_source_order · Uses: G001, S031

S033 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 0]

After

derived = [[3, 4, 0, 1, 2], [3, 4, 0, 1, 2], No]

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S032

S034 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 1]

After

derived = [[4, 0, 1, 2, 3], [4, 0, 1, 2, 3], No]

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S033

S035 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2]

After

derived = [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], Yes]

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S034

S036 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 3]

After

derived = [[1, 2, 3, 4, 0], [1, 2, 3, 4, 0], No]

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S035

S037 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 4]

After

derived = [[2, 3, 4, 0, 1], [2, 3, 4, 0, 1], No]

Conditions: none

The SAME e must satisfy e*x=x AND x*e=x for EVERY x.A one-sided condition or one row alone is insufficient.

Step identifiers and dependencies

Rule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S036

S038 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 0]

After

derived = [[[0, 3, 3], [1, 4, 4], [2, 0, 0], [3, 1, 1], [4, 2, 2]], [4]]

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S037

S039 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 1]

After

derived = [[[0, 4, 4], [1, 0, 0], [2, 1, 1], [3, 2, 2], [4, 3, 3]], [3]]

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S038

S040 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 2]

After

derived = [[[0, 0, 0], [1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]], [2]]

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S039

S041 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 3]

After

derived = [[[0, 1, 1], [1, 2, 2], [2, 3, 3], [3, 4, 4], [4, 0, 0]], [1]]

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S040

S042 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], 2, 4]

After

derived = [[[0, 2, 2], [1, 3, 3], [2, 4, 4], [3, 0, 0], [4, 1, 1]], [0]]

Conditions: none

Enumerate EVERY y for this x,and require BOTH products equal the SAME e.Retain all candidates;without established associativity,do not assume inverse uniqueness.

Step identifiers and dependencies

Rule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S041

S043 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]]]

After

derived = [[2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes]

Conditions: none

Without a unique two-sided e,the inverse-row list is empty and all-have-inverse is No,not vacuous True.An identity/inverse inventory still does not include associativity.

Step identifiers and dependencies

Rule: Group_unique_identity_required_NO_vacuous_inverse_claim · Uses: G001, S042

S044 · implies

Before

source roles = [5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]]]

After

derived = [Yes, Yes, [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes]

Conditions: none

Conjoin closure,ALL associativity triples,a two-sided identity and a two-sided inverse for EVERY element to classify THIS table as a group.No successful subset of obligations replaces the others.

Step identifiers and dependencies

Rule: Group_FULL_conjunction_all_finite_axiom_obligations · Uses: G001, S043

Reverse checks

  • S044 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S043 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S042 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S041 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S040 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S039 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S038 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S037 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S036 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S035 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S034 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S033 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S032 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S031 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S030 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S029 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S028 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S027 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S026 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S025 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S024 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S023 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S022 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S021 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S020 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S019 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S018 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S017 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S016 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S015 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S014 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S013 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S012 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S011 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S010 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S009 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S008 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S007 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S006 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S005 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S004 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S003 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S002 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
  • S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.

Result

(5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], Yes, Yes, [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 556d5ef24d5d0e630044f891df45a6fd3092b2227034e3e88c910354702cac01
  • candidate record SHA-256: 956acbeeba39165eded6244557fd65a2e17cac273fb7c56dce36508effeccf92
  • semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
  • question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON