### A direct factor theorem for commutative group algebras

Suppose $F$ is a field of characteristic $p\ne 0$ and $H$ is a $p$-primary abelian $A$-group. It is shown that $H$ is a direct factor of the group of units of the group algebra $FH$.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Suppose $F$ is a field of characteristic $p\ne 0$ and $H$ is a $p$-primary abelian $A$-group. It is shown that $H$ is a direct factor of the group of units of the group algebra $FH$.

Suppose $p$ is a prime number and $R$ is a commutative ring with unity of characteristic 0 in which $p$ is not a unit. Assume that $G$ and $H$ are $p$-primary abelian groups such that the respective group algebras $RG$ and $RH$ are $R$-isomorphic. Under certain restrictions on the ideal structure of $R$, it is shown that $G$ and $H$ are isomorphic.

In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by ${g}_{1},\cdots ,{g}_{k}$ with order ${n}_{1},\cdots ,{n}_{k}\in \{1,2,\cdots \}\cup \left\{\infty \right\}$, then $${\beta}_{1}^{\left(2\right)}\left(G\right)\le k-1-\sum _{i=1}^{k}\frac{1}{{n}_{i}}\phantom{\rule{0.166667em}{0ex}},$$ where ${\beta}_{1}^{\left(2\right)}\left(G\right)$ denotes the first ${\ell}^{2}$-Betti number of $G$. We also show that any $k$-generated group with ${\beta}_{1}^{\left(2\right)}\left(G\right)\ge k-1-\epsilon $ must have girth greater than or equal $1/\epsilon $.

Let $R$ be an associative ring with identity and let $J\left(R\right)$ denote the Jacobson radical of $R$. $R$ is said to be semilocal if $R/J\left(R\right)$ is Artinian. In this paper we give necessary and sufficient conditions for the group ring $RG$, where $G$ is an abelian group, to be semilocal.

Let G be a group, R an integral domain, and V G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen...