Concept guide
Two complete operation tables are given. (A) For G={0,1,2,3} with table T, prove that (G,*) is a finite group: closure, associativity for all 4³=64 triples, a two-sided identity and two-sided inverses. (B) K={0,1,2} with table ∘ is symmetric, has a two-sided identity and two-sided inverses for every element, yet violates associativity at the triple (1,1,2), so it is not a group; also show that all 19 triples containing 0 pass, so sampling would misjudge it.
Finite group
derivation-mko-finite-group-01
Concept guide
The explicit 24-entry table is a valid group action, but it is neither free nor transitive.
Finite group action
derivation-mko-finite-group-action-01
Concept guide
The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation.
Orbit of a finite group action
derivation-mko-finite-group-orbit-01
Concept guide
Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions.
Stabilizer of a finite group action
derivation-mko-finite-group-stabilizer-01
Concept guide
For f:C4 to C2, f(a)=a mod 2, the normal kernel is {0,2}; fibers are its cosets and f is not injective.
Kernel of a finite group homomorphism
derivation-mko-finite-group-homomorphism-kernel-01
Concept guide
The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate.
Conjugacy class in a finite group
derivation-mko-finite-group-conjugacy-class-01
Concept guide
Z(G)={e}, directly a normal subgroup; G is nonabelian and r is not central despite commuting with itself.
Center of a finite group
derivation-mko-finite-group-center-01