What is the role of generators between homomorphism of two cyclic group?
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Answer:
you only need to know the value of the homomorphism on a generator of the group. If the group is infinite cyclic then one always gets a homomorphism by defining its value on a generator and then extending inductively. So if a generates the group and f(a) is the value of the homomorphism on a then f(a n n) = f(a) n n. It makes no difference how you represent a power of the generator. If the group is finite cyclic this may not work. For instance you can not define a homomorphism of the cyclic group of order 2 into the cyclic group of order 3 that is non identically zero. The infinite cyclic group is an example of a free group. It is the free group on one generator. Free means that there are no non-trivial relations among the elements of the group. For such groups one always gets a homomorphism from its values on generators of the group. So take the free group on two generators a and b. this group is all finite words made of powers of a and b e.g. a − 4 −4b 15 15a 2 2b If one knows the value of the homomorphism on a and b then f extends to the whole group in the same way was in the infinite cyclic group. the kernel of f is called the set of relations on the quotient group. In fact any group may be thought of as a free group modulo the group of relations
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