1 GROUP HOMOMORPHISMS TonyWhelanMarch 2010NOTATION There are two common notations for groups & group homomorphisms (and I tend to prefer the second): Domain Co-Domain Set Law ofComposition Identity Set Law ofComposition Identity Course Notes :G G φ ′ → G ∗ e G′ ′ ∗ e′ Domain Co-Domain Set Law ofComposition Identity Set Law ofComposition Identity Common Alternative : G Hφ → G G ∗ G e HH∗ He DEFINITIONS A group homomorphism is a function : G Hφ → , where ( ) , G G ∗ and ( ) , HH∗ are groups (with the specified multiplications) such that: for all , f g G ∈ , ( ) ( ) ( ) HG f g f g φ φ ϕ ∗ = ∗ In words: if we multiply two elements of G then apply φ to the product {which is G f g G ∗ ∈ }, we get the same result as if we apply φ to each element separately, and then multiply the two values { ( ) fφ and ( ) g φ } in H{to get ( ) ( ) Hf g H φ ϕ ∗ ∈ }.
