This dissertation deals with three topics inside loop and quasigroup theory. First, as a continuation of the project started by David Stanovský and Petr Vojtĕchovský, we study the commutator of congruences defined by Freese and McKenzie in order to create a more pleasing, equivalent definition of the commutator inside of loops. Moreover, we show that the commutator can be characterized by the generators of the inner mapping group of the loop. We then translate these results to characterize the commutator of two normal subloops of any loop.
Second, we study automorphic loops with the desire to find more examples of small orders. Here we construct a family of automorphic loops, called quaternionic automorphic loops, which have order 2n for n ≥ 3, and prove several theorems about their structure. Although quaternionic automorphic loops are nonassociative, many of their properties are reminiscent of the generalized quaternion groups.
Lastly, we find varieties of quasigroups which are isotopic to commutative Moufang loops and prove their full characterization. Moreover, we define a new variety of quasigroups motivated by the semimedial quasigroups and show that they have an affine representation over commutative Moufang loops similar to the semimedial case proven by Kepka
