# zbMATH — the first resource for mathematics

Block separations and inclusions. (English) Zbl 1185.20009
Summary: We investigate the separation of characters by blocks at different primes and the inclusions of $$q$$-blocks in $$p$$-blocks (viewed as sets of characters), and use these notions to prove results on the structure of the corresponding groups. In particular, we provide a new criterion for the nilpotence of a finite group $$G$$ based on the separation by principal blocks, and we show that a condition on block unions has strong structural consequences.

##### MSC:
 20C20 Modular representations and characters 20D15 Finite nilpotent groups, $$p$$-groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure
GAP
Full Text:
##### References:
 [1] C. Bessenrodt, Addendum to [3] (2006) [2] Bessenrodt, C.; Malle, G.; Olsson, J.B., Separating characters by blocks, J. London math. soc., 73, 493-505, (2006) · Zbl 1096.20010 [3] Bessenrodt, C.; Navarro, G.; Olsson, J.B.; Tiep, P.H., On the navarro-willems conjecture for blocks of finite groups, J. pure appl. algebra, 208, 481-484, (2007) · Zbl 1113.20012 [4] Brandt, J., A lower bound for the number of irreducible characters in a block, J. algebra, 74, 2, 509-515, (1982) · Zbl 0478.20009 [5] Brauer, R., Blocks of characters and structure of finite groups, Bull. amer. math. soc., 1, 1, 21-62, (1979) · Zbl 0418.20006 [6] Conway, J.H.; Curtis, R.T.; Norton, S.P.; Parker, R.A.; Wilson, R.A., Atlas of finite groups, (1985), Clarendon Press Oxford · Zbl 0568.20001 [7] Gorenstein, D., Finite simple groups, (1982), Plenum Press New York · Zbl 0182.35402 [8] Nagao, H.; Tsushima, Y., Representations of finite groups, (1989), Academic Press [9] Navarro, G.; Willems, W., When is a p-block a q-block?, Proc. amer. math. soc., 125, 1589-1591, (1997) · Zbl 0870.20010 [10] Olsson, J.B.; Stanton, D., Block inclusions and cores of partitions, Aequationes math., 74, 90-110, (2007) · Zbl 1173.20010 [11] Schönert, M., GAP - groups, algorithms, and programming, Lehrstuhl D für Mathematik, rheinisch westfälische technische hochschule, (1995), Aachen Germany [12] Thompson, J.G., Normal p-complements and irreducible characters, J. algebra, 14, 129-134, (1970) · Zbl 0205.32606
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.