On Unit Groups of Real Group Algebras of p-Mixed Abelian Groups

One of the fundamental problem in commutative group theory is the determining the structure, up to isomorphism, of the unit group U(RG) of the group algebra RG of an abelian group G over the real number field R. Although this was done by Molloy in 1984 and Mollov-Nachev in 1997 when G is a p-group for some prime p, no complete solution has appeared in the literature yet. Nevertheless, combining principal known results due to Perlis-Walker, Berman-Bogdan, Karpilovsky, May, Molloy-Nachev and providing connecting arguments, a full resolution of the problem can be obtained in the case if G is p-mixed. It is the purpose of this brief note to present such a description of U(RG). This result extends the afore mentioned result of Mollov-Nachev.