# On the number of characters in the principal p-block

(Joint work with A. Schaeffer Fry and Carolina Vallejo) Let G be a finite group, let p be a prime number and let B be a p-block of G with defect group D. Studying the structure of D by means of the knowledge of some aspects of B is a main area in character theory of finite groups. Let k(B) be the number of irreducible characters in the p-block B. It is well-known that k(B)=1 if, and only if, D is trivial. It is also true that k(B)=2 if, and only if, D \cong C_2. For blocks B with k(B)=3 it is conjectured that D \cong C_3. In this talk we restrict our attention to the principal p-block of G, B_0(G), that is, the p-block containing the trivial character of G. It is well known that the defect groups of the principal block of G are exactly the Sylow p-subgroups of G. In this case it is even true that k(B_0(G))=3 if, and only if, D\cong C_3. Recently, Koshitani and Sakurai have shown that k(B_0(G))=4 implies that D \in {C_2 x C2, C_4, C_5}. In this work we go one step further and analyse the isomorphism classes of Sylow p-subgroups of groups G for which B_0(G) has exactly 5 irreducible characters. In particular we show that if k(B_0)=5, then D \in {C_5, C_7, D_8, Q_8}.

This talk is part of the Algebra seminar series.