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Question 10935: Kernel, Image and Injectivity of a Qualified Finite Homomorphism

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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R0020-10935 · q-20261013-advance-20261023-r03-subgroup-homomorphism-0085

Original question and givens

Use complete qualified finite group Cayley tables with carrier labels 0 through size−1 and the given row/column order. Every group axiom is a premise already checked on the whole table. Results concern these finite groups; a homomorphism need not be injective or an isomorphism. source size=6, source table=[[4, 2, 5, 0, 3, 1], [2, 0, 4, 1, 5, 3], [5, 4, 3, 2, 1, 0], [0, 1, 2, 3, 4, 5], [3, 5, 1, 4, 0, 2], [1, 3, 0, 5, 2, 4]], target size=3, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], mapping=[1, 2, 0, 0, 2, 1]. Require this map to preserve all source products. Give (source size,target size,mapping,source labels mapping to the target identity,sorted distinct image,injective?).

  • G001 — Use complete qualified finite group Cayley tables with carrier labels 0 through size−1 and the given row/column order. Every group axiom is a premise already checked on the whole table. Results concern these finite groups; a homomorphism need not be injective or an isomorphism. source size=6, source table=[[4, 2, 5, 0, 3, 1], [2, 0, 4, 1, 5, 3], [5, 4, 3, 2, 1, 0], [0, 1, 2, 3, 4, 5], [3, 5, 1, 4, 0, 2], [1, 3, 0, 5, 2, 4]], target size=3, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], mapping=[1, 2, 0, 0, 2, 1]. Require this map to preserve all source products. Give (source size,target size,mapping,source labels mapping to the target identity,sorted distinct image,injective?).

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

group=source;carrier Fin(6);complete Cayley table=[[4, 2, 5, 0, 3, 1], [2, 0, 4, 1, 5, 3], [5, 4, 3, 2, 1, 0], [0, 1, 2, 3, 4, 5], [3, 5, 1, 4, 0, 2], [1, 3, 0, 5, 2, 4]]

After

typed operation closure=Yes;all associativity failures=[]

Conditions: none

Type the complete operation table and check every associativity triple,not only cells used by a subset or map.These tasks require qualified finite groups.

Step identifiers and dependencies

Rule: complete_group_table_closure_associativity · Uses: G001

S002 · implies

Before

group=source;candidate identity=0;left products row=[4, 2, 5, 0, 3, 1];right products column=[4, 2, 5, 0, 3, 1]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S001

S003 · implies

Before

group=source;candidate identity=1;left products row=[2, 0, 4, 1, 5, 3];right products column=[2, 0, 4, 1, 5, 3]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S002

S004 · implies

Before

group=source;candidate identity=2;left products row=[5, 4, 3, 2, 1, 0];right products column=[5, 4, 3, 2, 1, 0]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S003

S005 · implies

Before

group=source;candidate identity=3;left products row=[0, 1, 2, 3, 4, 5];right products column=[0, 1, 2, 3, 4, 5]

After

both sides equal carrier labels=Yes

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S004

S006 · implies

Before

group=source;candidate identity=4;left products row=[3, 5, 1, 4, 0, 2];right products column=[3, 5, 1, 4, 0, 2]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S005

S007 · implies

Before

group=source;candidate identity=5;left products row=[1, 3, 0, 5, 2, 4];right products column=[1, 3, 0, 5, 2, 4]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S006

S008 · implies

Before

group=source;element=0;derived identity=3;two-sided inverse candidates=[4]

After

inverse=4

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S007

S009 · implies

Before

group=source;element=1;derived identity=3;two-sided inverse candidates=[5]

After

inverse=5

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S008

S010 · implies

Before

group=source;element=2;derived identity=3;two-sided inverse candidates=[2]

After

inverse=2

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S009

S011 · implies

Before

group=source;element=3;derived identity=3;two-sided inverse candidates=[3]

After

inverse=3

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S010

S012 · implies

Before

group=source;element=4;derived identity=3;two-sided inverse candidates=[0]

After

inverse=0

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S011

S013 · implies

Before

group=source;element=5;derived identity=3;two-sided inverse candidates=[1]

After

inverse=1

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S012

S014 · implies

Before

group=target;carrier Fin(3);complete Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]]

After

typed operation closure=Yes;all associativity failures=[]

Conditions: none

Type the complete operation table and check every associativity triple,not only cells used by a subset or map.These tasks require qualified finite groups.

Step identifiers and dependencies

Rule: complete_group_table_closure_associativity · Uses: G001, S013

S015 · implies

Before

group=target;candidate identity=0;left products row=[0, 1, 2];right products column=[0, 1, 2]

After

both sides equal carrier labels=Yes

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S014

S016 · implies

Before

group=target;candidate identity=1;left products row=[1, 2, 0];right products column=[1, 2, 0]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S015

S017 · implies

Before

group=target;candidate identity=2;left products row=[2, 0, 1];right products column=[2, 0, 1]

After

both sides equal carrier labels=No

Conditions: none

Both left and right multiplication must return every x.Do not infer identity from the diagonal or assume label0.

Step identifiers and dependencies

Rule: derive_two_sided_group_identity · Uses: G001, S016

S018 · implies

Before

group=target;element=0;derived identity=0;two-sided inverse candidates=[0]

After

inverse=0

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S017

S019 · implies

Before

group=target;element=1;derived identity=0;two-sided inverse candidates=[2]

After

inverse=2

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S018

S020 · implies

Before

group=target;element=2;derived identity=0;two-sided inverse candidates=[1]

After

inverse=1

Conditions: none

Find y with xy=e and yx=e using the actual identity.The group premises ensure uniqueness;no fixed label is every element's inverse.

Step identifiers and dependencies

Rule: derive_group_inverses_from_actual_identity · Uses: G001, S019

S021 · implies

Before

source size=6;target size=3;whole mapping=[1, 2, 0, 0, 2, 1]

After

all original mapping cells typed before product preservation

Conditions: none

Type the entire total mapping in original source-index order before checking all products.Do not fill missing values from partial correct data.

Step identifiers and dependencies

Rule: whole_group_mapping_type_guard · Uses: G001, S020

S022 · implies

Before

source pair=[0, 0];source product=4;mapped operands=[1, 1]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S021

S023 · implies

Before

source pair=[0, 1];source product=2;mapped operands=[1, 2]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S022

S024 · implies

Before

source pair=[0, 2];source product=5;mapped operands=[1, 0]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S023

S025 · implies

Before

source pair=[0, 3];source product=0;mapped operands=[1, 0]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S024

S026 · implies

Before

source pair=[0, 4];source product=3;mapped operands=[1, 2]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S025

S027 · implies

Before

source pair=[0, 5];source product=1;mapped operands=[1, 1]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S026

S028 · implies

Before

source pair=[1, 0];source product=2;mapped operands=[2, 1]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S027

S029 · implies

Before

source pair=[1, 1];source product=0;mapped operands=[2, 2]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S028

S030 · implies

Before

source pair=[1, 2];source product=4;mapped operands=[2, 0]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S029

S031 · implies

Before

source pair=[1, 3];source product=1;mapped operands=[2, 0]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S030

S032 · implies

Before

source pair=[1, 4];source product=5;mapped operands=[2, 2]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S031

S033 · implies

Before

source pair=[1, 5];source product=3;mapped operands=[2, 1]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S032

S034 · implies

Before

source pair=[2, 0];source product=5;mapped operands=[0, 1]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S033

S035 · implies

Before

source pair=[2, 1];source product=4;mapped operands=[0, 2]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S034

S036 · implies

Before

source pair=[2, 2];source product=3;mapped operands=[0, 0]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S035

S037 · implies

Before

source pair=[2, 3];source product=2;mapped operands=[0, 0]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S036

S038 · implies

Before

source pair=[2, 4];source product=1;mapped operands=[0, 2]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S037

S039 · implies

Before

source pair=[2, 5];source product=0;mapped operands=[0, 1]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S038

S040 · implies

Before

source pair=[3, 0];source product=0;mapped operands=[0, 1]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S039

S041 · implies

Before

source pair=[3, 1];source product=1;mapped operands=[0, 2]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S040

S042 · implies

Before

source pair=[3, 2];source product=2;mapped operands=[0, 0]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S041

S043 · implies

Before

source pair=[3, 3];source product=3;mapped operands=[0, 0]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S042

S044 · implies

Before

source pair=[3, 4];source product=4;mapped operands=[0, 2]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S043

S045 · implies

Before

source pair=[3, 5];source product=5;mapped operands=[0, 1]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S044

S046 · implies

Before

source pair=[4, 0];source product=3;mapped operands=[2, 1]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S045

S047 · implies

Before

source pair=[4, 1];source product=5;mapped operands=[2, 2]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S046

S048 · implies

Before

source pair=[4, 2];source product=1;mapped operands=[2, 0]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S047

S049 · implies

Before

source pair=[4, 3];source product=4;mapped operands=[2, 0]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S048

S050 · implies

Before

source pair=[4, 4];source product=0;mapped operands=[2, 2]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S049

S051 · implies

Before

source pair=[4, 5];source product=2;mapped operands=[2, 1]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S050

S052 · implies

Before

source pair=[5, 0];source product=1;mapped operands=[1, 1]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S051

S053 · implies

Before

source pair=[5, 1];source product=3;mapped operands=[1, 2]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S052

S054 · implies

Before

source pair=[5, 2];source product=0;mapped operands=[1, 0]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S053

S055 · implies

Before

source pair=[5, 3];source product=5;mapped operands=[1, 0]

After

f(source product)=1;target product=1;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S054

S056 · implies

Before

source pair=[5, 4];source product=2;mapped operands=[1, 2]

After

f(source product)=0;target product=0;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S055

S057 · implies

Before

source pair=[5, 5];source product=4;mapped operands=[1, 1]

After

f(source product)=2;target product=2;equal=Yes

Conditions: none

The left route multiplies in the source then maps;the right maps both operands then multiplies in the target.Retain every mismatch without inferring injectivity or isomorphism.

Step identifiers and dependencies

Rule: homomorphism_every_source_product_pair · Uses: G001, S056

S058 · implies

Before

all bad source ordered pairs=[]

After

homomorphism=Yes

Conditions: none

Preservation for all ordered source pairs is required.One failure supplies a counterexample;one success does not establish the universal condition.

Step identifiers and dependencies

Rule: whole_homomorphism_audit · Uses: G001, S057

S059 · implies

Before

qualified homomorphism=[1, 2, 0, 0, 2, 1];derived target identity=0

After

kernel source labels=[2, 3];image=[0, 1, 2];injective=No

Conditions: none

The kernel maps to the actual target identity;the image is the distinct mapping values.A finite total map is injective exactly when image size equals source size.A constant homomorphism need not be injective.This is a mathematical kernel,not a software-kernel receipt.

Step identifiers and dependencies

Rule: kernel_image_after_qualified_homomorphism · Uses: G001, S058

Reverse checks

  • S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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.

Result

(6, 3, [1, 2, 0, 0, 2, 1], [2, 3], [0, 1, 2], No)

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: 901c6e4b999e2ea377ac22c5b446b5e4d9e87f3ed0837bef0c1fe3d7b7cafe45
  • candidate record SHA-256: fe8fbd6915e303b70bfbf8ae1a220be7884f3c94e9795aae1c3296b2c159da46
  • semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f
  • question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON