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R0048-24982 · q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0082
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
G001 — Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).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.
Before
source roles = [kernel-trivial-injective-boundary]
After
derived = [[kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary]]
Conditions: none
List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [Kernel ORIGINAL SOURCE group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], 0]
After
derived = [9, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8]], Yes]
Conditions: none
Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables.
Rule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S001
Before
source roles = [Kernel ORIGINAL SOURCE group, 9, FOR EVERY x,y,z in0..n-1]
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], [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], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 729, Yes]
Conditions: none
Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups.
Rule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S002
Before
source roles = [Kernel ORIGINAL SOURCE group, 0]
After
derived = [[0, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 8, 0, 0], [2, 7, 0, 0], [3, 6, 0, 0], [4, 5, 0, 0], [5, 4, 0, 0], [6, 3, 0, 0], [7, 2, 0, 0], [8, 1, 0, 0]], Yes]
Conditions: none
Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises.
Rule: Group_ALL_two_sided_inverses · Uses: G001, S003
Before
source roles = [Kernel ORIGINAL TARGET group, 64, [[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], 0]
After
derived = [9, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8]], Yes]
Conditions: none
Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables.
Rule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S004
Before
source roles = [Kernel ORIGINAL TARGET group, 9, FOR EVERY x,y,z in0..n-1]
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], [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], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 729, Yes]
Conditions: none
Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups.
Rule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S005
Before
source roles = [Kernel ORIGINAL TARGET group, 0]
After
derived = [[0, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 8, 0, 0], [2, 7, 0, 0], [3, 6, 0, 0], [4, 5, 0, 0], [5, 4, 0, 0], [6, 3, 0, 0], [7, 2, 0, 0], [8, 1, 0, 0]], Yes]
Conditions: none
Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises.
Rule: Group_ALL_two_sided_inverses · Uses: G001, S006
Before
source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], [[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], [0, 1, 2, 3, 4, 5, 6, 7, 8]]
After
derived = [[[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [0, 7, 7, 7, Yes], [0, 8, 8, 8, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 7, 7, Yes], [1, 7, 8, 8, Yes], [1, 8, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 7, 7, Yes], [2, 6, 8, 8, Yes], [2, 7, 0, 0, Yes], [2, 8, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 7, 7, Yes], [3, 5, 8, 8, Yes], [3, 6, 0, 0, Yes], [3, 7, 1, 1, Yes], [3, 8, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 7, 7, Yes], [4, 4, 8, 8, Yes], [4, 5, 0, 0, Yes], [4, 6, 1, 1, Yes], [4, 7, 2, 2, Yes], [4, 8, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 7, 7, Yes], [5, 3, 8, 8, Yes], [5, 4, 0, 0, Yes], [5, 5, 1, 1, Yes], [5, 6, 2, 2, Yes], [5, 7, 3, 3, Yes], [5, 8, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 7, 7, Yes], [6, 2, 8, 8, Yes], [6, 3, 0, 0, Yes], [6, 4, 1, 1, Yes], [6, 5, 2, 2, Yes], [6, 6, 3, 3, Yes], [6, 7, 4, 4, Yes], [6, 8, 5, 5, Yes], [7, 0, 7, 7, Yes], [7, 1, 8, 8, Yes], [7, 2, 0, 0, Yes], [7, 3, 1, 1, Yes], [7, 4, 2, 2, Yes], [7, 5, 3, 3, Yes], [7, 6, 4, 4, Yes], [7, 7, 5, 5, Yes], [7, 8, 6, 6, Yes], [8, 0, 8, 8, Yes], [8, 1, 0, 0, Yes], [8, 2, 1, 1, Yes], [8, 3, 2, 2, Yes], [8, 4, 3, 3, Yes], [8, 5, 4, 4, Yes], [8, 6, 5, 5, Yes], [8, 7, 6, 6, Yes], [8, 8, 7, 7, Yes]], Yes, null]
Conditions: none
For these same source and target groups,check f(xy)=f(x)f(y) for every pair and retain the first failed row/null.A total map need not be a homomorphism.Failure blocks kernel theorems;an internal fallback is not the true identity preimage.
Rule: Kernel_complete_TOTAL_map_ALL_product_pairs_before_theorems · Uses: G001, S007
Before
source roles = [[0, 1, 2, 3, 4, 5, 6, 7, 8], 0, 0]
After
derived = [[0], 0, 0]
Conditions: none
The kernel is the complete preimage of the actual target identity.Cancel f(e)=f(e)^2 in the target group to obtain f(e)=e_target,so the true kernel is nonempty;do not assume identity0.
Rule: Kernel_actual_TARGET_identity_complete_preimage · Uses: G001, S008
Before
source roles = [Kernel ACTUAL identity preimage, [0], 0]
After
derived = [Yes, [[0, 0, 0, Yes]], Yes, [[0, 0, Yes]], Yes, Yes]
Conditions: none
H is nonempty and unique.Check identity membership,all products and every inverse separately;their conjunction qualifies a subgroup.Size divisibility or finiteness alone is insufficient.
Rule: Subgroup_identity_ALL_products_ALL_inverses_separate · Uses: G001, S009
Before
source roles = [Kernel ACTUAL identity preimage, 0, [0], 0]
After
derived = [[[0, 0, 0, 0]], [0], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S010
Before
source roles = [Kernel ACTUAL identity preimage, 1, [0], 8]
After
derived = [[[0, 1, 8, 0]], [0], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S011
Before
source roles = [Kernel ACTUAL identity preimage, 2, [0], 7]
After
derived = [[[0, 2, 7, 0]], [0], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S012
Before
source roles = [Kernel ACTUAL identity preimage, 3, [0], 6]
After
derived = [[[0, 3, 6, 0]], [0], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S013
Before
source roles = [Kernel ACTUAL identity preimage, 4, [0], 5]
After
derived = [[[0, 4, 5, 0]], [0], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S014
Before
source roles = [Kernel ACTUAL identity preimage, 5, [0], 4]
After
derived = [[[0, 5, 4, 0]], [0], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S015
Before
source roles = [Kernel ACTUAL identity preimage, 6, [0], 3]
After
derived = [[[0, 6, 3, 0]], [0], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S016
Before
source roles = [Kernel ACTUAL identity preimage, 7, [0], 2]
After
derived = [[[0, 7, 2, 0]], [0], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S017
Before
source roles = [Kernel ACTUAL identity preimage, 8, [0], 1]
After
derived = [[[0, 8, 1, 0]], [0], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S018
Before
source roles = [[0], Yes, 0]
After
derived = [f(kl)=f(k)f(l)=e, f(k^-1)=f(k)^-1=e, f(gkg^-1)=f(g)e*f(g)^-1=e]
Conditions: none
Products,inverses and every conjugate map to target identity,giving the general closure/normality argument alongside this complete finite audit.No quotient/isomorphism is claimed.
Rule: Kernel_GENERAL_closure_normality_from_homomorphism · Uses: G001, S019
Before
source roles = [0, [0], [0, 1, 2, 3, 4, 5, 6, 7, 8]]
After
derived = [[[0, [0]], [1, [1]], [2, [2]], [3, [3]], [4, [4]], [5, [5]], [6, [6]], [7, [7]], [8, [8]]], Yes, Yes, Yes, No]
Conditions: none
If f is injective,f(k)=f(e) forces k=e.If K={e},image equality gives a^-1b=e and a=b.The fixed No withholds quotient/first-isomorphism claims,not this iff or injectivity.
Rule: Kernel_trivial_actual_identity_iff_injective_NO_quotient_claim · Uses: G001, S020
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 4b8345d6dde030284d6f3cb8f87d1ed661fd0b7649d0b5b15d8132cf044aca32candidate record SHA-256: 830b663d3ca126c2e8d1971455ce0e5a2bb31634173cfa7ad3c9b702c3135169semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9Lean: NOT_RUN · human review: NOT_PERFORMED