Let f:G->H be a group homomorphism.
Prove or disprove the following statement.

1.Let a be an element of G. If f(a) is of finite order, then a is also of finite order.

2.Let f be a surjection. Then f is an isomorphism iff the order of the element f(a) is equal to the order of the element a , for all a belong to G.

First, I want to know if f is the trivial homorphism, then both will fail, right?

Second, if f is non-trivial homomorphism. Both will hold, right?

Finally, Please help me to prove both or give me counterexamples without considering the trivial homomorphism.

Solution Preview

1. False.
No matter f is a trivial or non-trivial group homomorphism, the statement is false. Here is a counter example. Let G=Z be the additive group of integers, H=Z2={0,1} is the additive group of Z mod 2. f:G->H is defined as f(a)=a mod 2 for all a in G. Then every nonzero element a in G has infinite order, but f(a) in H has order 0 or 1.

2. True.
No ...

Solution Summary

Homomorphisms and surjections are investigated. The solution is detailed and well presented.

Homomorphism
(a) Find the formulas for all group homomorphisms from Z_18 into Z_30.
(b) Choose one of the nonzero formulas in part (a), and for this formula find the kernel and image, and show how elements of the image correspond to costs of the kernel.

Suppose R is a ring, G, M are R-modules and Hom(G,M) is the set of R-module homomorphisms from G to M.
Identify Hom(Z/nZ,Z), Hom(Z,Z/nZ), Hom(Z/3Z,Z/6Z), Hom(Z/10Z,Z/6Z) as abelian groups, where n belongs to Z and Z is the set of integers.

Let @:G-->H be a homomorphism of G onto H, and let N be a normal subgroup of G. Show that @(N) is a normal subgroup of H.
How do I prove that a mapping is a normal subgroup of a group? What I am missing here is some understanding of the terminology and some clear understanding of mappings, homomorphisms, subgroups and no

Please attached for the question to help me understand Group and Ring Homomorphisms.
Is this function a Homomorphism?
a) Does 1:22 :22 where 1(a1,a2) = (a1,0) define a group homomorphism? a ring homomorphism? Prove your answers.
b) Does 1:22 :22

Define f: [0, 1) →C by f (x) = e2πix. Prove that f is one-one, onto, and continuous. Find a point x ∈ [0, 1) and a neighborhood N of x in [0, 1) such that f (N) is not a neighborhood of f (x) in C. Deduce that f is not a homomorphism.
See the attached file.

Problem:
Prove the Second Isomorphism Theorem: If A is an ideal of R and S is a subring of R, then S+A is a subring, A, and (S intersecting A) are ideals of S+A and S, respectively, and (S+A)/A isomorphic to A/(S intersecting A).

Let G = (Z/3Z)^4 SemiDirectProduct S_4 be the semi-direct product of (Z/3Z)^4 and S_4. Here S_4 acts on (Z/3Z)^4 by permutating the coordinates.
Hint: Given H1, H2 an element in (Z/3Z)^4 and K1, K2 an element in S4. The semi-direct product is given by the operation (H1, K1) * (H2, K2) = (H1 + K1(H2), K1 * K2)
A) Find the C