On isomorphism problem of the normalized unit group of the group algebras
by
Zsolt Balogh
University of Debrecen, Debrecen

The significancy of the isomorphism problem stimulates the study of the unit group. The isomorphism problem of the normalized unit group of the group algebras is the following. Let G and H be finite p-groups. Then the groups of the normalized units V(F_pG) and V(F_pH) are isomorphic if and only if G and H are isomorphic. Berman obtained the following result:

Let G and H be finite abelian p-groups. Then the groups of the normalized units V(F_pG) and V(F_pH) are isomorphic if and only if G and H are isomorphic.

We extended this result for field of p^m elements and finite p-groups of order pⁿ⁺¹ and exponent pⁿ.

Theorem. Let F_p^m be the field of p^m elements and let G and H be two finite p-groups of order pⁿ⁺¹ and exponent pⁿ. Then V(F_p^mG) and V(F_p^mH) are isomorphic if and only if G and H are isomorphic.

Date received: July 31, 2000