Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

SZAKÁCS, A., Unitary subgroup of the Sylow 2-subgroup of the group of normalized units in an infinite commutative group ring

Unitary subgroup of the Sylow 2-subgroup of the group of. 97 Let us consider the case of infinite ordinal A. Let A be an arbitrary infinite ordinal R = K\,H = G\ / ^A+I and the Sylow 2-subgroup S\ of the group G\ is not singular. Then W(KGf

X C W(RH) C V

2(RH) and by transfinite induction it is easy to prove the equation (2) V

2 (KG)

2" = V

2(RH). As compared to the group V

2(RH) we can construct the set M as in the above shown cases A), B) and C). Since in every of this cases the set M consist of the elements of the form x — y~

ly* and, by (2), y belongs to the group V

2(RH) = V

2(KGf , it follows that the elements x are the representatives of the cosets of group W

2 (KG)[

2] by the subgroup W

2 {KG)[

2]. Therefore for an arbitrary infinite ordinal A the Ulm-Kaplansky invariants of the group W(KG) can be calculated in the above shown way for the case A = n. References [1] A. A. BOVDI AND A. A. SZAKACS, The unitary subgroup of the group of units in a modular group algebra of a finite abelian p-group, Math. Zametki 6 45 (1989), 23-29 (hfRussian). (English translation Math. Notes

, 5-6 45 (1989), 445-450.) [2] A. SZAKACS, Unitary subgroup of the Sylow p-subgroup of the group of normalized units in an infinite commutative group ring. Acta Acad. Paed. Agriensis. Sec. Math. XXII (1994) 85-93. [3] A. A. BOVDI AND Z. F. PATAY, The structure of the centre of the multiplicative group of group ring of p-group over a ring of characteristic p. Vesci Akad. Nauk. Bssr. Ser. Fiz. Math. Nauk. (1978) No. 1, 5-11. ATTILA SZAKÁCS KÖRÖSI CSOMA SÁNDOR COLLEGE INSTITUTE OF BUSINESS FINANCE DEPARTMENT OF MATHEMATICS 5600 BÉKÉSCSABA, BAJZA U. 33. H-HUNGARY