Let G be a finite group, let p be a prime, and let P be a Sylow p-subgroup of G. For a subset S⊆ G, define operatornameIrr^S(G)=chi∈operatornameIrr(G)|chi(s)≠0 for some s∈ S. Say that P is TI when P^g cap P∈1,Pqquadfor all g∈ G. Alperin's Conjecture C. If P is TI, then |operatornameIrr^P(G)|=|operatornameIrr(N_G(P))|. The source states that Blau and Michler proved this in 1990 using the classification of finite simple groups, so the status is solved.

Alperin's Conjecture C

At least 10 years old · documented by 2015

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$ -subgroup of $G$ . For a subset $S\subseteq G$ , define $\operatorname{Irr}^S(G)=\{\chi\in\operatorname{Irr}(G)\mid\chi(s)\neq0\text{ for some }s\in S\}$ . Say that $P$ is TI when

P^g\cap P\in\{1,P\}\qquad\text{for all }g\in G.

Alperin's Conjecture C. If $P$ is TI, then

|\operatorname{Irr}^P(G)|=|\operatorname{Irr}(N_G(P))|.

The source states that Blau and Michler proved this in 1990 using the classification of finite simple groups, so the status is solved.

References

Primary source

Alexander Moretó, “The Main Problem of Block Theory: Picky Elements and Subnormalizers”, arXiv:2604.24565 (2026).

Additional references

7 papers in this index state this conjecture (2015–2026). The statement above is taken from the most recent of them; the others are arXiv:2208.12762, arXiv:2204.06373, arXiv:2204.00428, arXiv:2010.04499, arXiv:1804.06954, arXiv:1512.01145.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

The conjecture was proved in 1990, so this problem is settled rather than open.

Alperin’s Conjecture C asserts an equality between two character counts whenever a Sylow subgroup has the TI intersection property. Blau and Michler proved it in 1990 using the classification of finite simple groups.

Known results

Blau and Michler (1990): proved Conjecture C for finite groups with TI Sylow $p$ -subgroups, using the classification of finite simple groups.
Later work records Conjecture C for finite simple groups through explicit case analyses.

2026 developments

A 2026 paper revisits the TI Sylow-subgroup setting through the theory of picky elements and cites the Blau–Michler result; it reports no counterexample or gap affecting Conjecture C itself. Its additional counterexamples concern a stronger picky conjecture, not this character-counting statement.

Current status (as of August 2026): Conjecture C is settled by Blau and Michler’s 1990 proof; no unresolved case or credible challenge to that theorem was found.

Sources

No solutions have been posted yet.