A defect group of a block 
of a finite group algebra in characteristic 
is a maximal 
-subgroup 
for which the Brauer image of 
is nonzero. For a 
-subgroup 
, the Brauer map on 
retains the coefficients
of group elements centralizing 
, giving an element of 
. The defect groups of one block are conjugate, so their
order is well defined. The principal block, containing the trivial
module, has Sylow p-subgroups as defect
groups. A block of defect zero has trivial defect group. Donovan's
conjecture bounds the number of Morita equivalence
classes when this order is bounded.
Defect Group
See also
Block of a Group Algebra, Donovan Conjecture, Sylow p-SubgroupExplore with Wolfram|Alpha
References
Eaton, C. W.; Eisele, F.; and Livesey, M. "Donovan's Conjecture, Blocks with Abelian Defect Groups and Discrete Valuation Rings." Math. Z. 295, 249-264, 2020. https://doi.org/10.1007/s00209-019-02354-1.Webb, P. A Course in Finite Group Representation Theory. Cambridge, England: Cambridge University Press, 2016. https://doi.org/10.1017/CBO9781316677216.Cite this as:
Weisstein, Eric W. "Defect Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DefectGroup.html