If A and B are subsets of a group G, define
AB = {ab | a 2 A, b 2 B}. Now suppose phi: G -> G0 is a homomorphism of groups and N = Ker(phi) is its kernel.
(i) If H is a subgroup of G, show that HN = NH. (Warning: this is an equation of sets; proceed
accordingly; do not assume that G is abelian.)
(ii) Show that phi-inverse[phi[H]] = HN.
Thanks.
Solution Summary
Kernels and Homomorphisms are investigated. The solution is detailed and well presented.
Group Theory : Homomorphisms, Kernels, Isomorphisms and Fields. ... (b) If h: G1 ?> G2 is a homomorphism, define the kernel of h.Prove that the range of h is ...
Kernel of Phi and Homomorphism. Let phi is a homomorphism from Z30 onto a group order 3. Determine the kernel of phi. Find all generators of the kernel of phi. ...
... Є S | akk=0} is the kernel of f d ... C-->M2(R) a+bi -->[ab] is an interjecting homomorphism. ... Commutative Rings, Ideals, Kernels, Matrices and Injective and ...
... Definition: ker f = {g in G | f(g) = e}] But the kernel of a homomorphism is trivial (consists only of the identity element) if and only if it is injective. ...
... Show that the map : →/. =. is a homomorphism. What is the kernel of ? ... whence π is a group homomorphism. The kernel of π is H since π ( g ) = H iff g ∈ H . ...
(a) The kernel of this homomorphism is the principal ideal (x-1). Therefore, Z[x]/(x-1) is isomorphic to Z. According to the correspondence theorem, ideals of ...
... Consider the evaluation homomorphism E_g:F[x] -> E given by E_g (p(x)) = p(g) The image of E_g is F[g], the kernel of E_g, by the above, is m(x)F[x]. Now, Im ...
... This shows that θ is a homomorphism. ... To show (ii), recall first that the kernel of θ, Ker θ, is the subgroup of G that gets sent to the identity e, thus h ...
