Concept guide
Subgroups and homomorphisms: auditing on the complete tables of ℤ/4 and ℤ/2
G={0,1,2,3} with operation (a+b) mod 4 and K={0,1} with operation (a+b) mod 2 are given by complete Cayley tables. (1) Prove that H={0,2} is a subgroup of G and H'={0,1} is not; (2) prove that the mapping table f(0)=0, f(1)=1, f(2)=0, f(3)=1 is a homomorphism G→K, covering all 16 input pairs; (3) find the kernel and the image, and show that f is surjective but not injective, hence not an isomorphism; (4) show that the constant map h≡1 is not a homomorphism.