arXiv subjects

Victor Bovdi

Publications and source records attributed to Victor Bovdi.

23 records · Page 2Linked to original sources

arXiv2000

On elements in algebras having finite number of conjugates

Let $R$ be a ring with unity and $U(R)$ its group of units. Let $ΔU=\{a\in U(R)\mid [U(R):C_{U(R)}(a)]<\infty\}$ be the $FC$-radical of $U(R)$ and let $\nabla(R)=\{a\in R\mid [U(R):C_{U(R)}(a)]<\infty\}$ be the $FC$-subring of $R$. An infinite subgroup $H$ of $U(R)$ is said to be an $ω$-subgroup if the left annihilator of each nonzero Lie commmutator $[x,y]$ in $R$ contains only finite number of elements of the form $1-h$, where $x,y \in R$ and $h\in H$. In the case when $R$ is an algebra over a field $F$, and $U(R)$ contains an $ω$-subgroup, we describe its $FC$-subalgebra and the $FC$-radical. This paper is an extension of [1].

math.RA↗

arXiv2000

On the order of the unitary subgroup of modular group algebra

Let KG be a group algebra of a finite p-group G over a finite field K of characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group with a cyclic group of order 4 or G has an abelian subgroup A of index 2 and an element b such that b inverts each element of A.

math.RA↗

arXiv2000

Group algebras whose involutory units commute

Let K be a field of characteristic 2 and G a nonabelian locally finite 2-group. Let V(KG) be the group of units with augmentation 1 in the group algebra KG. An explicit list of groups is given, and it is proved that all involutions in V(KG) commute with each other if and only if G is isomorphic to one of the groups on this list. In particular, this property depends only on G and not at all on K.

math.RA↗

arXiv2000

On symmetric units in group algebras

Let $U(KG)$ be the group of units of the group ring $KG$ of the group $G$ over a commutative ring $K$. The anti-automorphism $g\mapsto g\m1$ of $G$ can be extended linearly to an anti-automorphism $a\mapsto a^*$ of $KG$. Let $S_*(KG)=\{x\in U(KG) \mid x^*=x\}$ be the set of all symmetric units of $U(KG)$. We consider the following question: for which groups $G$ and commutative rings $K$ it is true that $S_*(KG)$ is a subgroup in $U(KG)$. We answer this question when either a) $G$ is torsion and $K$ is a commutative $G$-favourable integral domain of characteristic $p\geq 0$ or b) $G$ is non-torsion nilpotent group and $KG$ is semiprime.

math.RA↗

arXiv2000

On a filtered multiplicative basis of group algebras

Let $K$ be a field of characteristic $p$ and $G$ a nonabelian metacyclic finite $p$-group. We give an explicit list of all metacyclic $p$-groups $G$, such that the group algebra $KG$ over a field of characteristic $p$ has a filtered multiplicative $K$-basis. We also present an example of a non-metacyclic 2-group $G$, such that the group algebra $KG$ over any field of characteristic 2 has a filtered multiplicative $K$-basis.

math.RA↗