Hölder's formula for the number of groups of a square-free order is an early advance in the enumeration of finite groups. This paper gives a structural proof of Hölder's result that is accessible to undergraduates. We introduce a number of group theoretic concepts such as nilpotency, the Fitting subgroup, and extensions. These topics, which are usually not covered in undergraduate group theory, feature in the proof of Hölder's result and have wide applicability in group theory. Finally, we remark on further results and conjectures in the enumeration of finite groups.
"Groups of a Square-Free Order,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 11:
1, Article 7.
Available at: https://scholar.rose-hulman.edu/rhumj/vol11/iss1/7