Today we will introduce four special kinds of homomorphisms.
1. An epimorphism f:G➔H maps the group G onto the group H, which means that for every element h of H, there is at least one element g of G that satisfies f(g) = h.
Example: f:(Z,+)➔(Z2,+) where f(even)=0 and f(odd)=1.
This is the group theory representation of the rules of adding even and odd numbers.
even + even = even
even + odd = odd + even = odd
odd + odd = even
Even is the identity element and odd is its own inverse.
2. A monomorphism f:G➔H maps every element of G to a unique element of H.
Example: f:(Z2,+)➔(Z4,+) where f(0)=0 and f(1)=2.
This is not an epimorphism, because nothing gets mapped to the elements 1 or 3.
3. An isomorphism f:G➔H is both an epimorphism and a monomorphism. As far as group theory is concerned, two isomorphic groups are identical in structure and there will also be an inverse map f⁻¹:H➔G, such that f⁻¹(f(g))=g for all g in G and f(f⁻¹(h))=h for all h in H.
If they are finite groups, they must be of the same order. With infinite groups, things can get tricky, but that's true any time infinity is involved.
Example #1: f:(Z,+)➔(2Z,+), where f(n)=2n for all n in Z. The inverse f⁻¹(2Z,+)➔(Z,+) is defined by f⁻¹(2n)=n.
It can easily be shown that {..., -4, -2, 0, 2, 4, ...} is a proper subset of {..., -4, -3, -2, -1, 0, 1, 2, 3, 4,...}, but they have the same order, both are countably infinite.
When Georg Cantor produced his work on the properties of the infinite, there was a lot of pushback, but eventually, mathematicians accepted his concepts, even though many feel paradoxical.
Example #2: f:(Z4,+)➔({1, i, -1, -i},✕), where f(n) = iⁿ.
The imaginary number i is defined by i²=-1. By extension, i³=-i and i⁴=1. We will look at other finite groups in terms of complex multiplication in future posts.
4. An automorphism f:G➔G is an isomorphism of a group onto itself.
The identity automorphism, where f(x) = x is always works, the group structure remains unchanged. But are there other automorphisms of a group? In many cases there are.
Example #1: f:(Z,+)➔(Z,+) defined as f(x)=-x for all x in Z.
The function f is its own inverse. If we can find a way to negate all elements of a additive group, then f(x)=-x will be an automorphism. It works just as well for (Q,+) and (R,+), which is to say, the rationals and the reals.
Example #2: f:(Z4,+)➔(Z4,+) defined as follows.
f(0)=0, f(1)=3,