For the following definitions, Let G and H be two groups: A morphism, rho, from G to H is a function rho: G --> H such that:
1)$ Rho $(1G) = 1H
2)Rho(g*gprime) = Rho(g)*Rho(gprime), this preserves the multiplication table
The domain and the codomain are two operations that are defined on every morphism. Morphims satisfy two axioms:
1)Associativity: h composed of (g composed of f) = (hcircleg)circlef whenever the operations are defined
2)Identity: for every object X, the identity morphism on X exists such that for every morphism f: A --> B,
idB composed f = f = f circle idA
Types of morphisms: An epimorphism is a morphism where for every h in H, there is at least one g in G with f(g) = h • This is the same as saying that rho is surjective or onto A monomorphism is a morphism for which rho(g) = rho(gprime) can only happen if g = gprime • This is the same as saying that rho is injective An isomorphism is a morphism that is both an epimorphism and a monomorphism (both surjective and injective). This means that rho sets up a 1-to-1 correspondence between the elements of G and the elements of H. • This is the same as saying that rho is bijective An automorphism is an isomorphism from a function to itself. It is a way of mapping the object to itself while preserving all of its structure. • An inner automorphism Is a function ƒ: G → G such that ƒ(x) = a−1xa, for all x in G, where a is a given fixed element of G. A homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). • Types of homomorphisms: o Group homomorphism- this is a homomorphism between two groups. o Ring homomorphism- this is a homomorphism between two rings. o Functor- this is a homomorphism between two categories o Linear map- this is a homomorphism between two vector spaces o Algebra homomorphism- this is a homomorphism between two algebras • Properties of elements under homomorphisms: Let phi be a homomorphism from a group G to a grou H and let g be and element of G. Then: 1) Phi carries the identity of G to the identity of H 2) Phi(g^n) = (phi(g))^n for all n in Z 3) If |g| is finite, then |phi(g)| divides |g| 4) Ker(phi) is a subgroup of G 5) aKer(phi) = bKern(phi) if and only if phi(a) = phi(b) 6) If phi(g) = gprime then phi^-1(gprime) = {x in G \ phi(x) = gprime} = gKerphi
• Properties of Subgroups Under Homomorphisms Let phi be a homomorphism from a group G to a group H and let I be a subgroup of G. Then: 1) Phi(I) = [phi(i) | i in I} is a subgroup of H 2) If I is cyclic, then phi(I) is cyclic 3) If I is Abelian, then phi(I) is Abelian 4) If I is normal in G, then phi(I) is normal in phi(G) 5) If \Kerphi\ = n, then phis is an n-to-1 mapping from G onto phi(G) 6) If |I| = n, then |phi(I)| divides n 7) If I bar is a subgroup of G bar, then phi^-1(I bar) = {i in G | phi(i) in Ibar} is a subgroup of G. 8) If I bar is a normal subgroup of G bar, then phi^-1(Ibar) = {i in G\ phi(i) in Ibar} is a normal subgroup of G 9) If phi is onto and Kerphi = {e}, then phi is an isomorphism from G to G bar. Examples • Any isomorphism is a homomorphism that is also onto and 1-to-1 • The mapping phi from Z to Zn, definded by phi(m) = m mod n is a homomorphism • The mapping phi(x) = x^2 from R*, the nonzero real numbers under multiplication, to itself is a homomorphism. This is because phi(ab) =(ab)^2 = a^2b^2 = phi(a)phi(b) for all a and b in R* • The exponential function rho : x e^x is an isomorphism. It is injective (monomorphism) and surjective (epimorphism) because one can take logs. • Square root: (R_t_, *) (R_t_, *) is an isomorphism • ( *2) : z/2z __> Z/3Z is a monomorphism,epimorphism and isomorphism
