Brauer's height-zero conjecture

About 8 years old · traced to 2018

Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a $p$ -block with defect group $D$ . The characters in

\operatorname{Irr}_0(B):=\{\chi\in\operatorname{Irr}(B)\mid \chi(1)_p=|G:D|_p\}

are the characters of height zero. Brauer's height-zero conjecture.

\operatorname{Irr}(B)=\operatorname{Irr}_0(B)\quad\Longleftrightarrow\quad D\text{ is abelian}.

The conjecture gives a character-theoretic criterion for abelian defect groups. The source notes proofs for $p$ -solvable groups and for 2-blocks with Sylow 2-subgroup defect, but does not state a general resolution.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Brauer's height zero conjecture
Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a $p$ -block of $G$ with defect group $D$ . An irreducible character in $B$ has height zero when its height is zero. Brauer's height zero conjecture. The defect group $D$ is abelian if and only if all irreducible characters in $B$ have height zero. This block-theoretic analogue of the Itô–Michler theorem relates the structure of a defect group to character heights; its resolution status is not established by the supplied source context.
source: Alexander Moretó, “The Main Problem of Block Theory: Picky Elements and Subnormalizers”, arXiv:2604.24565 (2026).

References

Primary source

Gunter Malle and Radha Kessar, “Local-global conjectures and blocks of simple groups”, arXiv:1804.06954 (2018).

Additional references

3 papers in this index state this conjecture (2015–2018). The statement above is taken from the most recent of them; the others are arXiv:1801.04272, arXiv:1512.01145.

Source: https://arxiv.org/abs/1804.06954 Brauer (1955), Brauer's height-zero conjecture

Posted 1 month ago

Brauer's height-zero conjecture

Brauer's height zero conjecture

Primary source

Additional references

Progress summary

Solutions 0