(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

(i) Every group-theoretic relation p=q satisfied by (a,b,c) in G a group is also satisfied by (x,y,z) in F a group.

(ii) There exists a homomorphism between G and F a->x b->y c->z.

Problem:

Show by example (i) can hold and (ii) cannot.

Show (i) can hold and (ii) can hold but not be a unique homomorphism.

3. The attempt at a solution

I'm having trouble trying to solve the first part of this question. Any help would be appreciated.

As for part 2.

Let G = (ZxZxZxZ, +) e_i is the identity in all spots except ith spot of the 4-tuple, 1 otherwise. EX: e_1 = (1,0,0,0)

a = e_1

b = e_2

c = e_3

Its pretty clear that any relation p satisfied by a,b,c must be equal to the form:

A_1(a) + A_2(b) + A_3(c) where A_i is the # of times you add or subtract.

Let F = (ZxZxZxZ, +)

x = e_1

b = e_2

c = e_3

e_4 -> e_4

Obviously this is one homomorphism were (i) and (ii) both hold

x = e_1

b = e_2

c = e_4

e_4 -> e_3

This is another homomorphism were (i) and (ii) both hold. Thus the homomorphism is not unique.

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# Homework Help: Group Homomorphisms

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