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.

Author Bio

Iordan Ganev is a Bulgarian-American attending Miami University as part of the class of 2010. He is pursuing a Bachelor of Science degree in mathematics and statistics and a second major in environmental science. During the academic year 2008-2009, Iordan studied at Royal Holloway, University of London. Under the supervision of Dr. Benjmain Klopsch at Royal Holloway, he completed a project that was the foundation for this paper. Iordan plans to obtain a doctorate in mathematics upon graduation. Iordan enjoys music, travel, archeology, and mountain hiking.