2019
Том 71
№ 11

# Bovdi A. A.

Articles: 6
Article (Ukrainian)

### Existence of a normal complement of a group in the multiplicative group of its group ring

Ukr. Mat. Zh. - 1991. - 43, № 7-8. - pp. 884–889

Article (Ukrainian)

### Generalized nilpotency of the multiplicative group of a group ring

Ukr. Mat. Zh. - 1989. - 41, № 9. - pp. 1179–1183

Article (Ukrainian)

### Group algebras with polycyclic multiplicative group

Ukr. Mat. Zh. - 1986. - 38, № 3. - pp. 373–375

Article (Ukrainian)

### Normal group rings

Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 3 – 8

Article (Ukrainian)

### Periodic normal subgroups of the multiplicative group of a group algebra

Ukr. Mat. Zh. - 1984. - 36, № 3. - pp. 282 - 286

Article (Russian)

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

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 136-141

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.