Verifying Huppert's Conjecture for the Simple Groups of Lie Type of Rank Two

In the late 1990s, Bertram Huppert posed a conjecture which, if true, would sharpen the connection between the structure of a nonabelian finite simple group H and the set of its characterdegrees. Specifically, Huppert made the following conjecture.

Huppert's Conjecture: Let G be a finite group and H a finite nonabelian simple group such that the sets of character degrees of G and H are the same. Then G is isomorphic to the direct product of H and A, where A is an abelian group.

To lend credibility to his conjecture, Huppert verified it on a case-by-case basis for many nonabelian simple groups, including the Suzuki groups, many of the sporadic simple groups, and a few of the simple groups of Lie type. Except for the Suzuki groups and the family of simple groups PSL₂(q), for q>3 prime or a power of a prime, Huppert proves the conjecture for specific simple groups of Lie type of small, fixed rank. We extend Huppert's results to all the linear, unitary, symplectic, and twisted Ree simple groups of Lie type of rank two.

In this dissertation, we will verify Huppert's Conjecture for ²G₂(q²) for q²=3^2m+1, m>0, and show progress toward the verification of Huppert's Conjecture for the simple group G₂(q) for q>4. We will also outline our extension of Huppert's results for the remaining families of simple groups of Lie type of rank two, namely PSL₃(q) for q>8, PSU₃(q²) for q>9, and PSp₄(q) for q>7.