LetRbe a commutative ring with identity of prime characteristicpand letGbe an abelian group. Denote byS(RG) the Sylowp-subgroup of the group of normalized units (i.e. the units of augmentation 1) of the group ringRG. The following result is established in the paper.1) LetSbe a subring ofRcontaining the identity ofR. Then the additive factor-groupR(p)/S(p)is isomorphic to the multiplicative groupR[p]/S[p], whereR(p)= {x∈R∣xp= 0} andR[p]= {y∈R∣yp= 1}.2) It holdsR(p)/Rp(p) ≌R[p]/Rp[p] whereRp= {xp∣x∈R}.Computing of the Ulm-Kaplansky invariants of the groupS(RG) is the main result of the paper. A description of the maximal divisible subgroup ofS(RG) is given as well