Similar formulas can be found for cubic and quartic equations. When mathematicians turned their attention to quintic equations, however, they hit a wall: they weren’t able to use previous techniques to find a “quintic formula”. Eventually, it was shown that this is because some quintic equations are not solvable by radicals. The method they used to show this is related to the following concept.
We have on occasion observed that different groups have the same Cayley table. We have also talked about different groups having the same structure. Regardless of whether a group of order two is additive or multiplicative, its elements behave in precisely the same fashion. The groups may look superficially different because of their features and operations, but the “group behaviour” is identical.
However, striking differences exist in the details. We want to study isomorphism of groups in quite a bit of detail, so to define isomorphism precisely, we start by reconsidering another topic that you studied in the past, functions. There we will also introduce the related notion of homomorphism. 21 This is the focus of Section 4.1. Section 4.2 lists some results that should help convince you that the existence of an isomorphism does show that two groups have an identical group structure. Section 4.3 describes how we can create new isomorphisms from a homomorphism’s kernel, a particular subgroup defined by a homomorphism. Section 4.4 introduces a class of isomorphism that is important for later applications, an automorphism.
The homomorphism property should remind you of certain special functions and operations that you have studied in Linear Algebra or Calculus. Recall from Exercise 2.26 that R+, the set of all positive real numbers, is a multiplicative group.
You might suspect that we only have to show that fA is a one-to-one, onto homomorphism, but this is not true. We have to show first that fA is well-defined and converter for scientific notation.