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, 3, 0, 5, 2, 1], [3, 0, 1, 4, 5, 2], [0, 1, 2, 3, 4, 5], [5, 4, 3, 2, 1, 0], [2, 5, 4, 1, 0, 3], [1, 2, 5, 0, 3, 4]], nonempty candidate subset=[0, 2, 4] normalized to sorted distinct labels. Give (size,subset,all products stay in the subset?).
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, 3, 0, 5, 2, 1], [3, 0, 1, 4, 5, 2], [0, 1, 2, 3, 4, 5], [5, 4, 3, 2, 1, 0], [2, 5, 4, 1, 0, 3], [1, 2, 5, 0, 3, 4]], nonempty candidate subset=[0, 2, 4] normalized to sorted distinct labels. Give (size,subset,all products stay in the subset?).
Step-by-step derivation
S001 · implies
Before
group=ambient;carrier Fin(6);complete Cayley table=[[4, 3, 0, 5, 2, 1], [3, 0, 1, 4, 5, 2], [0, 1, 2, 3, 4, 5], [5, 4, 3, 2, 1, 0], [2, 5, 4, 1, 0, 3], [1, 2, 5, 0, 3, 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, 3, 0, 5, 2, 1];right products column=[4, 3, 0, 5, 2, 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=[3, 0, 1, 4, 5, 2];right products column=[3, 0, 1, 4, 5, 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, S002
S004 · implies
Before
group=ambient;candidate identity=2;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, S003
S005 · implies
Before
group=ambient;candidate identity=3;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, S004
S006 · implies
Before
group=ambient;candidate identity=4;left products row=[2, 5, 4, 1, 0, 3];right products column=[2, 5, 4, 1, 0, 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, S005
S007 · implies
Before
group=ambient;candidate identity=5;left products row=[1, 2, 5, 0, 3, 4];right products column=[1, 2, 5, 0, 3, 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=2;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=2;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=2;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=2;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=2;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=2;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, 2, 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, 2, 4];ordered pair=[0, 2];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, 2, 4];ordered pair=[0, 4];ambient product=2
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, 2, 4];ordered pair=[2, 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, 2, 4];ordered pair=[2, 2];ambient product=2
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, 2, 4];ordered pair=[2, 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, 2, 4];ordered pair=[4, 0];ambient product=2
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, 2, 4];ordered pair=[4, 2];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, 2, 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
nonempty finite subset=[0, 2, 4];product closed=Yes
After
requested field records closure in this qualified finite group
Conditions: none
This field reports product closure.In a finite group,a nonempty closed subset does imply a subgroup:repeating powers supply identity and inverses.The separate task lists each obligation explicitly;do not falsely call closure insufficient in this finite setting.
Step identifiers and dependencies
Rule: finite_nonempty_closure_not_false_insufficiency_claim · Uses: G001, S022