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Pointers in creating a homomorphism

Finding a homomorphism between two groups is often difficult especially for a first course in group theory. There are many misconceptions and we address some of those here.

Between any two groups there is at least one homomorphism defined by tex2html_wrap_inline1831 . This is known as the trivial homomorphism. The set of homomorphisms between groups G and H is known as Hom(G,H), or in the case H=G, End(G). Thus, tex2html_wrap_inline1843 . We are often concerned with finding non trivial ones, if there are any.

For finite groups, we can make use of counting theorems. For example, the index of tex2html_wrap_inline1775 is a divisor of the order of G and is equal to the order of tex2html_wrap_inline1777 . So we could not construct an epimorphism from tex2html_wrap_inline851 to tex2html_wrap_inline1179 since 6 does not divide 8. Nor could we construct a homomorphism from tex2html_wrap_inline851 to tex2html_wrap_inline1179 such that the image is a cyclic subgroup of 3 elements, since tex2html_wrap_inline851 has no normal subgroup of index 3. The only choice would be that the image is one of the three cyclic subgroups of order 2 in tex2html_wrap_inline1179 . Then, the kernel of such a homomorphism would be a normal subgroup of order 4 in tex2html_wrap_inline851 , and there are 3 such subgroups. Another application shows that there are no non trivial homomorphisms from a cyclic group of order m to one of order n whenever m and n are coprime.

Keep in mind that the homomorphism is completely determined by the images of a generating set for the domain G. Also, domain elements can only map to elements in the codomain of order a divisor of the domain element. This again shows why no non trivial homomorphism from tex2html_wrap_inline851 to tex2html_wrap_inline1179 can have elements of order three in tex2html_wrap_inline1869 . But why can't all the elements of order 2 and the identity of tex2html_wrap_inline1179 be in the image? It's because they do not form a subgroup of tex2html_wrap_inline1179 . As another application, there is no monomorphism from tex2html_wrap_inline1875 to tex2html_wrap_inline1877 . As another, if tex2html_wrap_inline1879 is a homomorphism then the image of tex2html_wrap_inline1881 is either tex2html_wrap_inline1883 or an element of order 3.

If G is cyclic note that tex2html_wrap_inline1777 is also cyclic. To see this, let tex2html_wrap_inline1889 . Then there is a g in G with tex2html_wrap_inline1797 . As tex2html_wrap_inline1897 , say, then tex2html_wrap_inline1899 for some integer k. Then, tex2html_wrap_inline1903 . Hence tex2html_wrap_inline1777 is generated by tex2html_wrap_inline1907 showing the image of G is cyclic.

Similarly, it is easily shown that if G is abelian then tex2html_wrap_inline1777 is abelian. However, note that these two results do not say that H is cyclic, or H is abelian. So for an example, if you wanted to construct a homomorphism from tex2html_wrap_inline1919 then it could not be an epimorphism.

One final misconception is that the homomorphism must be onto. No, unless it is stated that it is an epimorphism (or onto) the image need only be a subgroup of the codomain.

next up previous
Next: Natural homomorphism Up: Homomorphisms Previous: Homomorphisms

Peter Williams
Sun Mar 30 14:48:35 PST 1997