# Number of blocks of characters of a finite group with a given defect

**Abstract**

R. Brauer's problem of the group-theoretic characteristic of the number of blocks with a given defect is considered in this paper. The following theorems are proved:

Theorem 1. The number of blocks with defect $d$ of a group $G$ does not exceed the number of $p$-regular non-nilpotent classes with defect $d$.

The theorem is a reinforcement of a result of Brauer — Nesbitt. An example is given showing that this estimate is not always attained.

Corollary 3. Let $G$ be a finite group of order $p^aq ((p, q) = 1$, $p$ is a prime number) containing a normal subgroup $H$ of the order $p^{\gamma}q (0 < \gamma < a)$, some sylow $p$-subgroup of which is a normal subgroup of $H$. Then the number of blocks with defect $d$ coincides with the number of $p$-regular non-nilpotent classes of $G$ with the same defect.

Theorem 3. There exist no blocks with zero defect in the group $G$ if, and only if, all classes with defect zero are nilpotent.

A new proof is also presented for Brauer's theorem on the number of blocks with a maximum defect.

**Citation Example:** *Bovdi A. A.* Number of blocks of characters of a finite group
with a given defect // Ukr. Mat. Zh. - 1961. - **13**, № 2. - pp. 136-141.

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