# Question 29263: Two-Sided Identity and Two-Sided Inverse Inventories

[Return to catalogue page 147](https://ocme.evemisslab.com/en/questions/page/147/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261022-r05-finite-group-foundations/#q-20261013-advance-20261022-r05-finite-group-0063)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0057-29263 · q-20261013-advance-20261022-r05-finite-group-0063



## Original question and givens

The complete finite Cayley table=[[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 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 closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.


- G001 — The complete finite Cayley table=[[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 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 closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.





## 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.


- [mko-finite-group](https://ocme.evemisslab.com/en/mko/mko-finite-group/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-str22-group-identity-inverse-inventory, group-identity-inverse-inventory]

After

derived = [[n, table, identities, inverse_rows, all_have_inverse]]

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 dependenciesRule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001





S002 · impliesBefore

source roles = [5, 0, [4, 2, 0, 1, 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 dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S001





S003 · impliesBefore

source roles = [5, 1, [2, 3, 1, 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 dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S002





S004 · impliesBefore

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 dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S003





S005 · impliesBefore

source roles = [5, 3, [1, 4, 3, 0, 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 dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S004





S006 · impliesBefore

source roles = [5, 4, [3, 0, 4, 2, 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 dependenciesRule: Group_EVERY_original_row_closure · Uses: G001, S005





S007 · impliesBefore

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

After

derived = [[4, 2, 0, 1, 3], [4, 2, 0, 1, 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 dependenciesRule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S006





S008 · impliesBefore

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

After

derived = [[2, 3, 1, 4, 0], [2, 3, 1, 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 dependenciesRule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S007





S009 · impliesBefore

source roles = [5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 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 dependenciesRule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S008





S010 · impliesBefore

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

After

derived = [[1, 4, 3, 0, 2], [1, 4, 3, 0, 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 dependenciesRule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S009





S011 · impliesBefore

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

After

derived = [[3, 0, 4, 2, 1], [3, 0, 4, 2, 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 dependenciesRule: Group_EACH_candidate_identity_BOTH_sides_ALL_elements · Uses: G001, S010





S012 · impliesBefore

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

After

derived = [[[0, 4, 4], [1, 2, 2], [2, 0, 0], [3, 1, 1], [4, 3, 3]], [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 dependenciesRule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S011





S013 · impliesBefore

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

After

derived = [[[0, 2, 2], [1, 3, 3], [2, 1, 1], [3, 4, 4], [4, 0, 0]], [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 dependenciesRule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S012





S014 · impliesBefore

source roles = [5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 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 dependenciesRule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S013





S015 · impliesBefore

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

After

derived = [[[0, 1, 1], [1, 4, 4], [2, 3, 3], [3, 0, 0], [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 dependenciesRule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S014





S016 · impliesBefore

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

After

derived = [[[0, 3, 3], [1, 0, 0], [2, 4, 4], [3, 2, 2], [4, 1, 1]], [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 dependenciesRule: Group_EVERY_element_ALL_two_sided_inverse_candidates · Uses: G001, S015





S017 · impliesBefore

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

After

derived = [[2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], 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 dependenciesRule: Group_unique_identity_required_NO_vacuous_inverse_claim · Uses: G001, S016









## Reverse checks


- 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, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]], [2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], 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: 694483e5ee0394473bad0e2014758fdbe780c6448c7229a90ec889de5cd19cde
- candidate record SHA-256: 91fb312c0b9e62a0f1da891a3242660e09fd62658a85ed1203b240ff38308c25
- semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
- question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r05-finite-group-0063.json)