Recognising Ree Groups Using the Codegree Set

Abstract

The codegree of an irreducible character $\chi $ of a finite group G is $|G : \ker \chi |/\chi (1)$ . We show that the Ree group ${}^2G_2(q)$ , where $q = 3^{2f+1}$ , is determined up to isomorphism by its set of codegrees.

On the Characterization of Sporadic Simple Groups by Codegrees

“…and PSL (3,3) by [1,4,10]. The conjecture has also been verified for PSL (3, q) and PSU(3, q) in [17] and 2 G 2 (q) in [11]. Most of these results concern simple groups with less than 21 character degrees [3].…”

On the Characterization of Alternating Groups by Codegrees

“…It has been verified for PSL(2, q), PSL (3,4), Alt 7 , J 1 , 2 B 2 (2 2f +1 ) where f ≥ 1, M 11 , M 12 , M 22 , M 23 and PSL (3,3) by [1,3,13]. The conjecture has also been verified for PSL (3, q) and PSU (3, q) in [19] and 2 G 2 (q) in [14]. Recently, the authors verified the conjecture for all sporadic simple groups in [11].…”

On the Characterisation of Sporadic Simple Groups by Codegrees

“…Using the order formulae for the 17 infinite families of nonabelian simple groups [6], we first deduce which simple groups have order dividing 448345497600. In this case, we get the following list: A 5 , A 6 , A 7 , A 8 , A 9 , A 10 , A 11 , A 12 , A 13 , PSL 2 (7), PSL 2 (11), PSL 2 (13), PSL 2 (25), PSL 2 (27), PSL 2 (64), PSL 3 (3), PSL 3 (4), PSL 3 (9), PSL 4 (2), PSL 4 (3), O 5 (2) , (2) . By Lemma 2.4, if a simple group K has cod(K) ⊆ cod(Suz), then it must belong to the above list.…”

“…6 , A 7 , A 8 , PSL 2(7), PSL 2(8), PSL 2(11), PSL 2 (23), PSL 2 (29), PSL 2 (31), PSL 2 (32), PSL 3 (4), PSL 5 (2), O 5 (2) , G 2 (2) , PSU 3 (3) Co 3 A 5 , A 6 , A 7 , A 8 , A 9 , A 10 , A 11 , A 12 , PSL 2 (7), PSL 2 (8), PSL 2 (11), PSL 2 (23), PSL 3 (4), O 5 (2) , O 5 (3), O 7 (2), G 2 (2) , PSU 3 (3) Co 2 A 5 , A 6 , A 7 , A 8 , A 9 , A 10 , A 11 , A 12 , PSL 2 (7), PSL 2 (8), PSL 2 (11), PSL 2 (23), PSL 3 (4), O 5 (2) , O 5 (3), O 7 (2), O + 8 (2), G 2 (2) , PSU 3 (3) Co 1 A 5 , A 6 , A 7 , A 8 , A 9 , A 10 , A 11 , A 12 , A 13 , A 14 , A 15 , A 16 , PSL 2 (7), PSL 2 (8), PSL 2 (11), PSL 2 (13), PSL 2 (23), PSL 2 (25), PSL 2 (27), PSL 2 (49), PSL 2 (64), PSL 3…”

“…A 6 , A 7 , A 8 , A 9 , A 10 , A 11 , A 12 , A 13 , PSL 2(7), PSL 2 (8), PSL 2(11), PSL 2 (13), PSL 2 (25), PSL 2 (27), PSL 2 (64), PSL 3 (3), PSL 3 (4), PSL 3(9), PSL 4 (3), O 5 (2) , O 5 (3), O 7 (2), O 7 (3), PSp 6 (3), O + 8 (2), G 2 (2) , G 2 (3), G 2 (4), PSU 3 (3), 2 B 2 (8), 2 F 4 (2) Fi 23 A 5 , A 6 , A 7 , A 8 , A 9 , A 10 , A 11 , A 12 , A 13 , PSL 2 (7), PSL 2 (8), PSL 2 (11), PSL 2 (13), PSL 2 (16), PSL 2 (17), PSL 2 (23), PSL 2 (25), PSL 2 (27), PSL 2 (64), PSL 3 (3), PSL 3 (4), PSL 3 (9), PSL 3 (16), PSL 4 (3), PSL…”