Abstract
In abstract algebra, the symmetric group Sn on a finite set X (containing n elements) is the set of all possible mappings from X to itself, and whose group operation is function composition. In practice, Sn = all possible permutations of set X’s n distinct elements; thus the magnitude of Sn is n!. We can partition Sn into one or more parts (also known as blocks or cells) – that is to say, we can divide Sn into a union of non-overlapping, non-empty subsets; these blocks form a basis for Sn. We can also say that they form a subalgebra of Sn if any combination of these parts via multiplication produces a linear combination of the blocks formed by the initial partitioning. This paper seeks to find out, for any given n, how many of the potential Bell number of ways to partition Sn actually yield a working subalgbera. Once all of the subalgebras for n = 1,...,2 are found, the intention is to try to discern if there are any mathematical patterns and/or categories that these partitions fall into. The number of possible ways to partition Sn increases dramatically as n increases, meaning that even for n = 4 we are already looking at a 15-digit number of possible ways to divide Sn. This means that checking all possible partitions of Sn via brute force would be prohibitively inefficient. Instead, this paper will attempt to find all possible subalgebras via several different approaches, including Monte Carlo simulations, as well as examining subsets of prohibitively large search spaces.
Advisor
Visa, Sofia
Department
Computer Science
Recommended Citation
Nelson, Peter C., "Needles and Haystacks: A Computational Approach to Finding Subalgebras of the Group Algebra of the Symmetric Group" (2016). Senior Independent Study Theses. Paper 13566.
https://openworks.wooster.edu/independentstudy/13566
Publication Date
2016
Degree Granted
Bachelor of Arts
Document Type
Senior Independent Study Thesis
© Copyright 2016 Peter C. Nelson