Blocks with 𝑝-power character degrees

Navarro, Gabriel; Robinson, Geoffrey

doi:10.1090/S0002-9939-05-07915-3

Blocks with $p$-power character degrees
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by Gabriel Navarro and Geoffrey R. Robinson PDF

Proc. Amer. Math. Soc. 133 (2005), 2845-2851 Request permission

Abstract:

Let $B$ be a $p$-block of a finite group $G$. If $\chi (1)$ is a $p$-power for all $\chi \in \operatorname {Irr}(B)$, then $B$ is nilpotent.

References

Antal Balog, Christine Bessenrodt, Jørn B. Olsson, and Ken Ono, Prime power degree representations of the symmetric and alternating groups, J. London Math. Soc. (2) 64 (2001), no. 2, 344–356. MR 1853455, DOI 10.1112/S0024610701002356
Richard Brauer, Investigations on group characters, Ann. of Math. (2) 42 (1941), 936–958. MR 5731, DOI 10.2307/1968775
Michel Broué and Lluís Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), no. 2, 117–128. MR 558864, DOI 10.1007/BF01392547
I. M. Isaacs and Stephen D. Smith, A note on groups of $p$-length $1$, J. Algebra 38 (1976), no. 2, 531–535. MR 393215, DOI 10.1016/0021-8693(76)90236-2
Burkhard Külshammer and Lluís Puig, Extensions of nilpotent blocks, Invent. Math. 102 (1990), no. 1, 17–71. MR 1069239, DOI 10.1007/BF01233419
G. Malle and A. E. Zalesskiĭ, Prime power degree representations of quasi-simple groups, Arch. Math. (Basel) 77 (2001), no. 6, 461–468. MR 1879049, DOI 10.1007/PL00000518
G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, vol. 250, Cambridge University Press, Cambridge, 1998. MR 1632299, DOI 10.1017/CBO9780511526015
John G. Thompson, Normal $p$-complements and irreducible characters, J. Algebra 14 (1970), 129–134. MR 252499, DOI 10.1016/0021-8693(70)90116-X
Atumi Watanabe, On nilpotent blocks of finite groups, J. Algebra 163 (1994), no. 1, 128–134. MR 1257309, DOI 10.1006/jabr.1994.1008