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. size=6, 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]], nonempty subset=[0, 3, 4] normalized as a set. Give (size,subset,group identity,product closed?,contains identity?,inverse closed?,all subgroup checks?).
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. size=6, 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]], nonempty subset=[0, 3, 4] normalized as a set. Give (size,subset,group identity,product closed?,contains identity?,inverse closed?,all subgroup checks?).
Step-by-step derivation
S001 · implies
Before
group=ambient;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=ambient;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=ambient;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=ambient;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=ambient;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=ambient;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=ambient;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=ambient;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=ambient;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=ambient;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=ambient;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=ambient;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=ambient;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
subset=[0, 3, 4];ordered pair=[0, 0];ambient product=4
After
product stays in subset=Yes
Conditions: none
Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required.
Step identifiers and dependencies
Rule: subset_every_ordered_product_closure · Uses: G001, S013
S015 · implies
Before
subset=[0, 3, 4];ordered pair=[0, 3];ambient product=0
After
product stays in subset=Yes
Conditions: none
Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required.
Step identifiers and dependencies
Rule: subset_every_ordered_product_closure · Uses: G001, S014
S016 · implies
Before
subset=[0, 3, 4];ordered pair=[0, 4];ambient product=3
After
product stays in subset=Yes
Conditions: none
Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required.
Step identifiers and dependencies
Rule: subset_every_ordered_product_closure · Uses: G001, S015
S017 · implies
Before
subset=[0, 3, 4];ordered pair=[3, 0];ambient product=0
After
product stays in subset=Yes
Conditions: none
Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required.
Step identifiers and dependencies
Rule: subset_every_ordered_product_closure · Uses: G001, S016
S018 · implies
Before
subset=[0, 3, 4];ordered pair=[3, 3];ambient product=3
After
product stays in subset=Yes
Conditions: none
Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required.
Step identifiers and dependencies
Rule: subset_every_ordered_product_closure · Uses: G001, S017
S019 · implies
Before
subset=[0, 3, 4];ordered pair=[3, 4];ambient product=4
After
product stays in subset=Yes
Conditions: none
Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required.
Step identifiers and dependencies
Rule: subset_every_ordered_product_closure · Uses: G001, S018
S020 · implies
Before
subset=[0, 3, 4];ordered pair=[4, 0];ambient product=3
After
product stays in subset=Yes
Conditions: none
Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required.
Step identifiers and dependencies
Rule: subset_every_ordered_product_closure · Uses: G001, S019
S021 · implies
Before
subset=[0, 3, 4];ordered pair=[4, 3];ambient product=4
After
product stays in subset=Yes
Conditions: none
Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required.
Step identifiers and dependencies
Rule: subset_every_ordered_product_closure · Uses: G001, S020
S022 · implies
Before
subset=[0, 3, 4];ordered pair=[4, 4];ambient product=0
After
product stays in subset=Yes
Conditions: none
Check every ordered subset pair using the ambient row/column roles.The group is finite and the candidate nonempty as required.
Step identifiers and dependencies
Rule: subset_every_ordered_product_closure · Uses: G001, S021
S023 · implies
Before
element=0;ambient inverse=4;subset=[0, 3, 4]
After
inverse retained=Yes
Conditions: none
Check membership of each inverse derived from the full group,not the element itself or a negated label.
Step identifiers and dependencies
Rule: subgroup_each_actual_inverse_membership · Uses: G001, S022
S024 · implies
Before
element=3;ambient inverse=3;subset=[0, 3, 4]
After
inverse retained=Yes
Conditions: none
Check membership of each inverse derived from the full group,not the element itself or a negated label.
Step identifiers and dependencies
Rule: subgroup_each_actual_inverse_membership · Uses: G001, S023
S025 · implies
Before
element=4;ambient inverse=0;subset=[0, 3, 4]
After
inverse retained=Yes
Conditions: none
Check membership of each inverse derived from the full group,not the element itself or a negated label.
Step identifiers and dependencies
Rule: subgroup_each_actual_inverse_membership · Uses: G001, S024
S026 · implies
Before
closed=Yes;actual identity=3;identity retained=Yes;inverse closed=Yes
After
all explicit subgroup obligations=Yes
Conditions: none
Retain the protocol's three explicit closure,identity and inverse judgments,then take their conjunction.
Step identifiers and dependencies
Rule: subgroup_explicit_joint_audit · Uses: G001, S025