Question 25781: The Induced Map Must Agree on Every Kernel-Coset Representative

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0050-25781 · q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0031

Original question and givens

Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 3, 1, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).

G001 — Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], identity=0, and the aligned total mapping=[0, 3, 1, 4, 2]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (kernel, cosets, representative_image_rows, well_defined, certificate).

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-first-isomorphism-theorem-finite-groups

Step-by-step derivation

S001 · implies

Before

source roles = [iso1-induced-well-definedness]

After

derived = [[kernel, cosets, representative_image_rows, well_defined, certificate]]

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 = [Iso1 ORIGINAL SOURCE, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], 0]

After

derived = [5, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4]], 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 = [Iso1 ORIGINAL SOURCE, 5, 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]], 125, [0, 4, 3, 2, 1]]

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 = [Iso1 ORIGINAL TARGET, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], 0]

After

derived = [5, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4]], 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, S003

S005 · implies

Before

source roles = [Iso1 ORIGINAL TARGET, 5, 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]], 125, [0, 4, 3, 2, 1]]

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, S004

S006 · implies

Before

source roles = [Iso1 ACTUAL kernel, [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]], 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, S005

S007 · implies

Before

source roles = [Iso1 ACTUAL image, [0, 1, 2, 3, 4], 0]

After

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

Conditions: none

Step identifiers and dependencies

Rule: Subgroup_actual_identity_product_inverse_ALL_conjugates · Uses: G001, S006

S008 · implies

Before

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

After

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

Conditions: none

Audit every homomorphism pair and the full target-identity preimage/image.Qualification allows a bad map and reports actual properties:homFalse can coexist with normal kernel or subgroup image.An empty kernel is not replaced by an identity fallback.

Step identifiers and dependencies

Rule: Iso1_ALL_product_laws_ACTUAL_kernel_image_properties_before_theorem · Uses: G001, S007

S009 · implies

Before

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

After

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

Conditions: none

Kernel normality gives(ah)(bk)=ab(b^-1hb)k in abK;check every representative product agrees on its quotient cell.Every f(g) in a coset must also agree before defining bar_f.The carrier consists of coset SETS,not least representatives.

Step identifiers and dependencies

Rule: Iso1_full_LEFT_coset_quotient_ALL_products_and_representative_images · Uses: G001, S008

S010 · implies

Before

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

After

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

Conditions: none

The identity f(ab)=f(a)f(b) preserves products.If f(a)=f(b),a^-1b is in K and the cosets agree,giving injectivity.Every image value f(g) is hit by gK,giving surjectivity onto the IMAGE.Equal cardinalities do not replace these obligations or give surjectivity to the whole target.

Step identifiers and dependencies

Rule: Iso1_ALL_induced_product_pairs_injective_and_onto_IMAGE_separate · Uses: G001, S009

Reverse checks

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.

Result

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

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: f464ee0bbe6449718c995294b48b47c23b9c8b54d4ae465f39cb44455f82e34e
candidate record SHA-256: 5e486c605843b1c982df20a1c1464a9a82a8a62f67500514db8bcd7f6a716001
semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82
question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f
Lean: NOT_RUN · human review: NOT_PERFORMED

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