This thesis was concerned with classifying the real indecomposable solvableLie algebras with codimension one nilradicals of dimensions two through seven.This thesis was organized into three chapters. In the first, we described the necessary concepts and definitions about Liealgebras as well as a few helpful theorems that are necessary to understand theproject. We also reviewed many concepts from linear algebra that are essentialto the research. The second chapter was occupied with a description of how we went aboutclassifying the Lie algebras. In particular, it outlined the basic premise ofthe classification: that we can use the automorphisms of the nilradical of theLie algebra to find a basis with the simplest structure equations possible. Inaddition, it outlined a few other methods that also helped find this basis.Finally, this chapter included a discussion of the canonical forms of certaintypes of matrices that arose in the project. The third chapter presented a sample of the classification of the sevendimensional Lie algebras. In it, we proceeded step-by-step through theclassification of the Lie algebras whose nilradical was one of fourspecifically chosen because they were representative of the different typesthat arose during the project. In the appendices, we presented our results in a list of the multiplicationtables of the isomorphism classes found.
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