02223873

j

02223873 Kobayashi, Yuji Undecidability of the centers of groups and group algebras. Arch. Math. 85, No. 3, 227-232 (2005). 2005 Birkh\"auser Verlag (Springer), Basel EN group algebras word problem algorithms centers finitely presented groups monoid algebras

doi:10.1007/s00013-005-1235-z

The main result of the article says that for any fixed integer $k>2$ (or $k=\infty$) and a field $K$, there exists a recursive family $\{G_n\}_{n\in\bbfN}$ of finitely presented groups with trivial centers such that each $G_n$ has solvable word problem, the dimension $d(n)$ of the center of the group algebra $KG_n$ equals either $k$ or $1$, and it is undecidable for a given $n$ whether $d(n)=1$ or $d(n)=k$. The author also discusses other (un)decidability questions related to the centers of groups, monoids and the centers of their group (monoid) algebras. Michael Dokuchaev (Murcia)