Simple groups with few irreducible character degrees

Abstract

In this paper, the non-abelian simple groups having at most 20 distinct irreducible character degrees are classified.

Recognising Ree Groups Using the Codegree Set

“…LEMMA 2.2. If a simple group G is isomorphic to PSL(2, k), 2 B 2 (q 2 ), PSL (3,4), Alt 7 , J 1 , M 11 , PSL (3,3), G 2 (2) , M 22 , PSL(4, 2), M 12 , M 23 , PSL(3, q), PSU(3, q) or 2 G 2 (q), then cod(G) can be found in Table 1.…”

“…{1, (q − 1)(q 2 + 1), q 2 (q − 1), 2 3 f +2 (q 2 + 1), q 2 (q − 2r + 1), q 2 (q + 2r + 1)} PSL (3,4) {1,…”

“…1 is the only nontrivial odd codegree, it must be the same as either 3 5 f +3 (q 2 − q + 1) or q 3 (q + 1 − 3m) or q 3…”

“…This implies that q 3 ≤ (q + 2 + 3m) 2 + 1, which is a contradiction. Suppose G/N PSL (3,4). Note that 3 2 •5•7 is the only nontrivial odd codegree in cod(G/N).…”

“…This conjecture was shown to hold for PSL (2, q) in [4]. In [1], the conjecture was proven for 2 B 2 (2 2 f +1 ), where f ≥ 1, PSL (3,4), Alt 7 and J 1 . The conjecture also holds in the cases where H is M 11 , M 12 , M 22 , M 23 or PSL (3,3) by [9].…”