1. The Herzog-Schönheim Conjecture for Finite Pyramidal Groups
- Author
-
Andaloro, Leah E.
- Subjects
- Mathematics, group, finite, pyramidal, Herzog, Schönheim, mathematics, conjecture, group theory, Herzog-Schönheim, finite pyramidal, finite pyramidal group, number theory, combinatorics, supersolvable, supersolvable groups, coset, coset partition, distinct indices, Mirsky-Newman Theorem, Burshtein’s Conjecture, covering system, exact covering system
- Abstract
In 1974, Marcel Herzog and Jochanan Schönheim proposed “Research Problem no. 9” in The Canadian Mathematical Bulletin. The conjecture states that for a group G with non-trivial, finite left coset partition P = {g_iG_i}_{i=1}^k, the indices [G : G_1],[G : G_2],...,[G : G_k] cannot be distinct. This conjecture is referred to as the Herzog-Schönheim Conjecture. In this paper, we will discuss how the conjecture holds for various finite groups. We will also examine the work of Marc Berger, Alexander Felzenbaum, and Aviezri Fraenkel in exploring the conjecture’s relationship to finite pyramidal groups.
- Published
- 2023