A note on central idempotents in group rings

Sonia P. Coelho
1987 Proceedings of the Edinburgh Mathematical Society  
Let G be a group and K a field. We denote by <2l(KG) the group of units of the group ring of G over K and for a group X we denote by T(X) the set of torsion elements of G, i.e., the set of all elements of finite order. In the study of group-theoretical properties of 'ft(KG) it has been found that some conditions on this group lead to the fact that T= T(G) is a group and every idempotent in KT is central in KG. For example, if K is a field of characteristic p>0, this will happen when G is
more » ... en when G is infinite, non abelian and <%(KG) is an FC-group ([1], Theorem A) or when G is nilpotent or FC, contains no p-elements and T(%(KG)) forms a subgroup ([4], Theorem 4.1). This condition has also appeared in the description of the structure of some unit groups, as in [5], Lemma VI.3.22. In this note, we shall study this condition and determine what it will imply about the structure of the original group'and the given field. We will consider only groups whose torsion elements form a locally finite group. This is the case in all the situations which were mentioned above. We shall assume that K is a field of characteristic p>0 and denote by ^(K) the prime subfield of K. Also, A will be the set of all p'-elements of G and P the set of all p-elements of G. We prove the following: Theorem. Let K be a field of characteristic p > 0 and G a group such that the torsion T of G forms a locally finite subgroup. Then, every idempotent of KT is central in KG if and only if: (i) A is an abelian subgroup of G. (ii) if A is non central, then the algebraic closure Q of &(K) in K is finite and for all xeG and all teA, there exists reN such that xtx~1 = t^. Furthermore, for each such r, we have that (Cl:&>{K))\r, (iii) P is a subgroup of G, (iv) T=PxA. Proof. Necessity. To establish (i), let teG be a p'-element. Then, e = o(t)~l(l + t+ ••• +t°l t)~l ) is a central idempotent. Given xeG, we have that xex~1 = e. By •This paper forms part of the
doi:10.1017/s0013091500017971 fatcat:bnuv7oy5ijcirj4qrvabhtxr3a