On the Brauer-Feit bound for abelian defect groups

Abstract

We improve the Brauer-Feit bound on the number of irreducible characters in a $p$-block for abelian defect groups by making use of Halasi and Podoski (Every coprime linear group admits a base of size two. http://arxiv.org/abs/1212.0199v1, [7]) and Kessar and Malle (Ann Math 178(2):321–384, [11]). We also prove Brauer’s $k(B)$-Conjecture for 2-blocks with abelian defect groups of rank at most 5 and 3-blocks and 5-blocks with abelian defect groups of rank at most 3.

Appendix

The following table is needed in Lemma 6.

Size	Id	Size	Id	Size	Id	Size	Id	Size	Id	Size	Id	Size	Id	Size	Id
8	3	48	7	64	95	64	174	64	251	80	36	96	79	96	136
12	4	48	14	64	97	64	176	64	253	80	37	96	80	96	137
16	7	48	15	64	98	64	177	64	254	80	38	96	81	96	138
16	8	48	17	64	99	64	178	64	255	80	39	96	82	96	139
16	11	48	25	64	101	64	186	64	258	80	40	96	83	96	144
16	13	48	29	64	115	64	187	64	261	80	41	96	87	96	145
20	4	48	33	64	116	64	189	64	263	80	42	96	88	96	146
21	1	48	35	64	117	64	196	64	265	80	44	96	89	96	147
24	5	48	36	64	118	64	198	72	5	80	46	96	90	96	148
24	6	48	37	64	119	64	201	72	6	80	50	96	91	96	149
24	8	48	38	64	121	64	202	72	8	80	51	96	92	96	153
24	14	48	39	64	123	64	203	72	17	81	7	96	93	96	154
28	3	48	40	64	124	64	205	72	20	84	8	96	98	96	155
32	9	48	41	64	128	64	206	72	21	84	12	96	99	96	156
32	11	48	43	64	129	64	207	72	22	84	13	96	100	96	157
32	19	48	47	64	130	64	210	72	23	84	14	96	101	96	158
32	25	48	48	64	131	64	211	72	25	88	5	96	102	96	160
32	27	48	51	64	133	64	213	72	27	88	7	96	103	96	168
32	28	52	4	64	134	64	215	72	28	88	9	96	104	96	179
32	30	56	4	64	137	64	216	72	30	93	1	96	105	96	186
32	31	56	5	64	138	64	217	72	32	96	4	96	106	96	187
32	34	56	7	64	140	64	218	72	33	96	5	96	107	96	189
32	39	56	9	64	141	64	219	72	35	96	6	96	108	96	192
32	40	56	12	64	142	64	220	72	46	96	7	96	109	96	195
32	42	60	12	64	144	64	221	72	48	96	12	96	110	96	200
32	43	63	3	64	145	64	223	72	49	96	13	96	111	96	206
32	46	64	6	64	146	64	226	76	3	96	16	96	113	96	207
32	48	64	8	64	147	64	227	80	4	96	27	96	114	96	208
32	50	64	10	64	149	64	228	80	5	96	28	96	115	96	209
36	4	64	12	64	150	64	229	80	6	96	30	96	116	96	210
36	10	64	32	64	152	64	230	80	7	96	32	96	117	96	211
36	12	64	34	64	155	64	231	80	14	96	33	96	118	96	212
36	13	64	38	64	157	64	232	80	15	96	34	96	119	96	213
40	5	64	41	64	159	64	233	80	16	96	35	96	120	96	214
40	6	64	52	64	161	64	234	80	17	96	44	96	121	96	215
40	8	64	67	64	162	64	235	80	25	96	54	96	122	96	216
40	10	64	71	64	163	64	236	80	26	96	61	96	123	96	217
40	12	64	73	64	167	64	237	80	28	96	62	96	124	96	219
40	13	64	75	64	169	64	240	80	29	96	64	96	125	96	223
48	4	64	89	64	170	64	243	80	30	96	67	96	126	96	226
48	5	64	90	64	171	64	244	80	31	96	68	96	134	96	230
48	6	64	92	64	173	64	250	80	34	96	78	96	135