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← Question derivation index · Original question batch
R0048-24955 · q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0030
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. 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, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
G001 — Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. 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, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).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-normal-subgroup-audit]
After
derived = [[kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate]]
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, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], 0]
After
derived = [12, [[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], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11]], 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, 12, 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, 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]], 1728, 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, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 11, 0, 0], [2, 10, 0, 0], [3, 9, 0, 0], [4, 8, 0, 0], [5, 7, 0, 0], [6, 6, 0, 0], [7, 5, 0, 0], [8, 4, 0, 0], [9, 3, 0, 0], [10, 2, 0, 0], [11, 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], [1, 0]], 0]
After
derived = [2, [[0, 0, 0, 0], [1, 1, 1, 1]], 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, 2, FOR EVERY x,y,z in0..n-1]
After
derived = [[[Yes, Yes], [Yes, Yes]], 8, 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, 1], [[0, 0, 0, 0], [1, 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, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], [[0, 1], [1, 0]], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]]
After
derived = [[[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 0, 0, Yes], [0, 3, 1, 1, Yes], [0, 4, 0, 0, Yes], [0, 5, 1, 1, Yes], [0, 6, 0, 0, Yes], [0, 7, 1, 1, Yes], [0, 8, 0, 0, Yes], [0, 9, 1, 1, Yes], [0, 10, 0, 0, Yes], [0, 11, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [1, 4, 1, 1, Yes], [1, 5, 0, 0, Yes], [1, 6, 1, 1, Yes], [1, 7, 0, 0, Yes], [1, 8, 1, 1, Yes], [1, 9, 0, 0, Yes], [1, 10, 1, 1, Yes], [1, 11, 0, 0, Yes], [2, 0, 0, 0, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [2, 4, 0, 0, Yes], [2, 5, 1, 1, Yes], [2, 6, 0, 0, Yes], [2, 7, 1, 1, Yes], [2, 8, 0, 0, Yes], [2, 9, 1, 1, Yes], [2, 10, 0, 0, Yes], [2, 11, 1, 1, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 0, 0, Yes], [3, 4, 1, 1, Yes], [3, 5, 0, 0, Yes], [3, 6, 1, 1, Yes], [3, 7, 0, 0, Yes], [3, 8, 1, 1, Yes], [3, 9, 0, 0, Yes], [3, 10, 1, 1, Yes], [3, 11, 0, 0, Yes], [4, 0, 0, 0, Yes], [4, 1, 1, 1, Yes], [4, 2, 0, 0, Yes], [4, 3, 1, 1, Yes], [4, 4, 0, 0, Yes], [4, 5, 1, 1, Yes], [4, 6, 0, 0, Yes], [4, 7, 1, 1, Yes], [4, 8, 0, 0, Yes], [4, 9, 1, 1, Yes], [4, 10, 0, 0, Yes], [4, 11, 1, 1, Yes], [5, 0, 1, 1, Yes], [5, 1, 0, 0, Yes], [5, 2, 1, 1, Yes], [5, 3, 0, 0, Yes], [5, 4, 1, 1, Yes], [5, 5, 0, 0, Yes], [5, 6, 1, 1, Yes], [5, 7, 0, 0, Yes], [5, 8, 1, 1, Yes], [5, 9, 0, 0, Yes], [5, 10, 1, 1, Yes], [5, 11, 0, 0, Yes], [6, 0, 0, 0, Yes], [6, 1, 1, 1, Yes], [6, 2, 0, 0, Yes], [6, 3, 1, 1, Yes], [6, 4, 0, 0, Yes], [6, 5, 1, 1, Yes], [6, 6, 0, 0, Yes], [6, 7, 1, 1, Yes], [6, 8, 0, 0, Yes], [6, 9, 1, 1, Yes], [6, 10, 0, 0, Yes], [6, 11, 1, 1, Yes], [7, 0, 1, 1, Yes], [7, 1, 0, 0, Yes], [7, 2, 1, 1, Yes], [7, 3, 0, 0, Yes], [7, 4, 1, 1, Yes], [7, 5, 0, 0, Yes], [7, 6, 1, 1, Yes], [7, 7, 0, 0, Yes], [7, 8, 1, 1, Yes], [7, 9, 0, 0, Yes], [7, 10, 1, 1, Yes], [7, 11, 0, 0, Yes], [8, 0, 0, 0, Yes], [8, 1, 1, 1, Yes], [8, 2, 0, 0, Yes], [8, 3, 1, 1, Yes], [8, 4, 0, 0, Yes], [8, 5, 1, 1, Yes], [8, 6, 0, 0, Yes], [8, 7, 1, 1, Yes], [8, 8, 0, 0, Yes], [8, 9, 1, 1, Yes], [8, 10, 0, 0, Yes], [8, 11, 1, 1, Yes], [9, 0, 1, 1, Yes], [9, 1, 0, 0, Yes], [9, 2, 1, 1, Yes], [9, 3, 0, 0, Yes], [9, 4, 1, 1, Yes], [9, 5, 0, 0, Yes], [9, 6, 1, 1, Yes], [9, 7, 0, 0, Yes], [9, 8, 1, 1, Yes], [9, 9, 0, 0, Yes], [9, 10, 1, 1, Yes], [9, 11, 0, 0, Yes], [10, 0, 0, 0, Yes], [10, 1, 1, 1, Yes], [10, 2, 0, 0, Yes], [10, 3, 1, 1, Yes], [10, 4, 0, 0, Yes], [10, 5, 1, 1, Yes], [10, 6, 0, 0, Yes], [10, 7, 1, 1, Yes], [10, 8, 0, 0, Yes], [10, 9, 1, 1, Yes], [10, 10, 0, 0, Yes], [10, 11, 1, 1, Yes], [11, 0, 1, 1, Yes], [11, 1, 0, 0, Yes], [11, 2, 1, 1, Yes], [11, 3, 0, 0, Yes], [11, 4, 1, 1, Yes], [11, 5, 0, 0, Yes], [11, 6, 1, 1, Yes], [11, 7, 0, 0, Yes], [11, 8, 1, 1, Yes], [11, 9, 0, 0, Yes], [11, 10, 1, 1, Yes], [11, 11, 0, 0, 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, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], 0, 0]
After
derived = [[0, 2, 4, 6, 8, 10], 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, 2, 4, 6, 8, 10], 0]
After
derived = [Yes, [[0, 0, 0, Yes], [0, 2, 2, Yes], [0, 4, 4, Yes], [0, 6, 6, Yes], [0, 8, 8, Yes], [0, 10, 10, Yes], [2, 0, 2, Yes], [2, 2, 4, Yes], [2, 4, 6, Yes], [2, 6, 8, Yes], [2, 8, 10, Yes], [2, 10, 0, Yes], [4, 0, 4, Yes], [4, 2, 6, Yes], [4, 4, 8, Yes], [4, 6, 10, Yes], [4, 8, 0, Yes], [4, 10, 2, Yes], [6, 0, 6, Yes], [6, 2, 8, Yes], [6, 4, 10, Yes], [6, 6, 0, Yes], [6, 8, 2, Yes], [6, 10, 4, Yes], [8, 0, 8, Yes], [8, 2, 10, Yes], [8, 4, 0, Yes], [8, 6, 2, Yes], [8, 8, 4, Yes], [8, 10, 6, Yes], [10, 0, 10, Yes], [10, 2, 0, Yes], [10, 4, 2, Yes], [10, 6, 4, Yes], [10, 8, 6, Yes], [10, 10, 8, Yes]], Yes, [[0, 0, Yes], [2, 10, Yes], [4, 8, Yes], [6, 6, Yes], [8, 4, Yes], [10, 2, 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, 2, 4, 6, 8, 10], 0]
After
derived = [[[0, 0, 0, 0], [2, 2, 0, 2], [4, 4, 0, 4], [6, 6, 0, 6], [8, 8, 0, 8], [10, 10, 0, 10]], [0, 2, 4, 6, 8, 10], 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, 2, 4, 6, 8, 10], 11]
After
derived = [[[0, 1, 11, 0], [2, 3, 11, 2], [4, 5, 11, 4], [6, 7, 11, 6], [8, 9, 11, 8], [10, 11, 11, 10]], [0, 2, 4, 6, 8, 10], 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, 2, 4, 6, 8, 10], 10]
After
derived = [[[0, 2, 10, 0], [2, 4, 10, 2], [4, 6, 10, 4], [6, 8, 10, 6], [8, 10, 10, 8], [10, 0, 10, 10]], [0, 2, 4, 6, 8, 10], 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, 2, 4, 6, 8, 10], 9]
After
derived = [[[0, 3, 9, 0], [2, 5, 9, 2], [4, 7, 9, 4], [6, 9, 9, 6], [8, 11, 9, 8], [10, 1, 9, 10]], [0, 2, 4, 6, 8, 10], 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, 2, 4, 6, 8, 10], 8]
After
derived = [[[0, 4, 8, 0], [2, 6, 8, 2], [4, 8, 8, 4], [6, 10, 8, 6], [8, 0, 8, 8], [10, 2, 8, 10]], [0, 2, 4, 6, 8, 10], 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, 2, 4, 6, 8, 10], 7]
After
derived = [[[0, 5, 7, 0], [2, 7, 7, 2], [4, 9, 7, 4], [6, 11, 7, 6], [8, 1, 7, 8], [10, 3, 7, 10]], [0, 2, 4, 6, 8, 10], 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, 2, 4, 6, 8, 10], 6]
After
derived = [[[0, 6, 6, 0], [2, 8, 6, 2], [4, 10, 6, 4], [6, 0, 6, 6], [8, 2, 6, 8], [10, 4, 6, 10]], [0, 2, 4, 6, 8, 10], 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, 4, 6, 8, 10], 5]
After
derived = [[[0, 7, 5, 0], [2, 9, 5, 2], [4, 11, 5, 4], [6, 1, 5, 6], [8, 3, 5, 8], [10, 5, 5, 10]], [0, 2, 4, 6, 8, 10], 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, 2, 4, 6, 8, 10], 4]
After
derived = [[[0, 8, 4, 0], [2, 10, 4, 2], [4, 0, 4, 4], [6, 2, 4, 6], [8, 4, 4, 8], [10, 6, 4, 10]], [0, 2, 4, 6, 8, 10], 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 = [Kernel ACTUAL identity preimage, 9, [0, 2, 4, 6, 8, 10], 3]
After
derived = [[[0, 9, 3, 0], [2, 11, 3, 2], [4, 1, 3, 4], [6, 3, 3, 6], [8, 5, 3, 8], [10, 7, 3, 10]], [0, 2, 4, 6, 8, 10], 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, S019
Before
source roles = [Kernel ACTUAL identity preimage, 10, [0, 2, 4, 6, 8, 10], 2]
After
derived = [[[0, 10, 2, 0], [2, 0, 2, 2], [4, 2, 2, 4], [6, 4, 2, 6], [8, 6, 2, 8], [10, 8, 2, 10]], [0, 2, 4, 6, 8, 10], 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, S020
Before
source roles = [Kernel ACTUAL identity preimage, 11, [0, 2, 4, 6, 8, 10], 1]
After
derived = [[[0, 11, 1, 0], [2, 1, 1, 2], [4, 3, 1, 4], [6, 5, 1, 6], [8, 7, 1, 8], [10, 9, 1, 10]], [0, 2, 4, 6, 8, 10], 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, S021
Before
source roles = [[0, 2, 4, 6, 8, 10], 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, S022
S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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, 2, 4, 6, 8, 10], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10], Yes], [1, [0, 2, 4, 6, 8, 10], Yes], [2, [0, 2, 4, 6, 8, 10], Yes], [3, [0, 2, 4, 6, 8, 10], Yes], [4, [0, 2, 4, 6, 8, 10], Yes], [5, [0, 2, 4, 6, 8, 10], Yes], [6, [0, 2, 4, 6, 8, 10], Yes], [7, [0, 2, 4, 6, 8, 10], Yes], [8, [0, 2, 4, 6, 8, 10], Yes], [9, [0, 2, 4, 6, 8, 10], Yes], [10, [0, 2, 4, 6, 8, 10], Yes], [11, [0, 2, 4, 6, 8, 10], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
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: 02307eda6cab5d8677d3e445c06bb0e26edc6cbb72838e23bd894366272e4521candidate record SHA-256: 934e88bba60a72d093625be2ee613c7b119e2abe8fa21bd4da8ca53dc58d657fsemantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9Lean: NOT_RUN · human review: NOT_PERFORMED