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Principal block has sylow defect

Statement

The principal block of $k G$ , namely the unique block acting as one on the trivial module $k$ , has Sylow $p$ -subgroups as its defect groups.

Facts & Assumptions

Given: A finite group in characteristic $p$ .

[F1]

A module vertex is conjugate into the block defect group. (Vertices of modules in a block lie in a defect group)

[F2]

Relative projectivity is equivalent to a trace equal to the identity. (Higman's criterion characterizes relative projectivity through the relative trace idempotent test)

[F3]

Every $p$ -subgroup lies in a conjugate of a Sylow subgroup. (Sylow II: in a finite group every $p$ -subgroup lies in a conjugate of any Sylow $p$ -subgroup, and the Sylow $p$ -subgroups form a single conjugacy class)

Proof

technique · direct

1.1

On the one-dimensional trivial module every endomorphism is a scalar. Its trace from $H$ is multiplication by $[G : H]$ times that scalar. Thus Higman says it is relatively $H$ -projective exactly when $p ∤ [G : H]$ . Among $p$ -subgroups this holds exactly for Sylow subgroups, and no proper subgroup of a Sylow has the property. Its vertices are therefore Sylow subgroups.

F2F3

2.1

The orthogonal central block idempotents act as orthogonal idempotent scalars summing to one, so exactly one acts as one on $k$ ; this is the principal block. By [F1] a Sylow vertex is conjugate into a defect group $D$ . Since $D$ is a $p$ -subgroup, it cannot strictly contain a Sylow subgroup, so it is Sylow. Conversely every Sylow is conjugate to $D$ by [F3], and conjugating the diagonal vertex keeps it a vertex of the same block.

F1F3step 1.1∎

Sources

Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245. Local argument and conventions as displayed above.

Depends on

Used by

Centralizer containment is sufficient but not necessary Counterexample
Blocks and defect groups of s3 Example
Brauer correspondence for a defect-D8 block of S7 in characteristic 2 Example
Defect zero and p group boundaries Example

Dependency tree · two levels

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Sources

Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245 (standard reference, not scraped)