TY - JOUR

T1 - Exponent and p-rank of finite p-groups and applications

AU - Sambale, Benjamin

N1 - Funding Information: Proof. By Proposition 4.13 in [9] we may assume that D is non-abelian. Since D has exponent at most 9, it follows that D =∼ C9 C3 or C9 C9. In the first case, Theorem 4.5 in [17] implies that all Cartan invariants of B are divisible by 3. The same holds in case |D| = 34 by Corollary 5 in [15] (cf. [17, Section 2]). Now Proposition 4.6 in [9] gives a contradiction. The last statement follows from Propositions 4.13 and 4.14 in [9]. □ Acknowledgements. This work is supported by the Carl Zeiss Foundation and the Daimler and Benz Foundation. The author thanks Heiko Dietrich, Bettina Eick, Max Horn, and Eamonn O’Brien for the assistance with computations of automorphism groups.

PY - 2014/7

Y1 - 2014/7

N2 - We bound the order of a finite p-group in terms of its exponent and p-rank. Here the p-rank is the maximal rank of an abelian subgroup. These results are applied to defect groups of p-blocks of finite groups with given Loewy length. Doing so, we improve results in a recent paper by Koshitani, Külshammer, and Sambale. In particular, we determine possible defect groups for blocks with Loewy length 4.

AB - We bound the order of a finite p-group in terms of its exponent and p-rank. Here the p-rank is the maximal rank of an abelian subgroup. These results are applied to defect groups of p-blocks of finite groups with given Loewy length. Doing so, we improve results in a recent paper by Koshitani, Külshammer, and Sambale. In particular, we determine possible defect groups for blocks with Loewy length 4.

KW - Exponent

KW - Loewy length

KW - p-rank

UR - http://www.scopus.com/inward/record.url?scp=84905487501&partnerID=8YFLogxK

U2 - 10.1007/s00013-014-0665-x

DO - 10.1007/s00013-014-0665-x

M3 - Article

AN - SCOPUS:84905487501

VL - 103

SP - 11

EP - 20

JO - Archiv der Mathematik

JF - Archiv der Mathematik

SN - 0003-889X

IS - 1

ER -