{'fraction_non_alphanumeric': 0.045998739760554505, 'fraction_numerical': 0.008191556395715185, 'mean_word_length': 4.533101045296167, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
ON THE BALANCED SUBGROUPS OF MODULAR GROUP RINGS

Let S(RG) be the normed p-unit group in a group ring RG, formed by an abelian group G and a commutative ring R with identity of prime characteristic p. All unexplained symbols and letters as well as the terminology and definitions from the abelian group theory (including the topological ones) can be found in the classical book monographs [7]. For a background material in that direction, we refer the reader also to [1]–[6]. The major goal motivating the present paper is to find some special nice and isotype subgroups of S(RG), a problem that arises naturally in the examination of the total projectivity both in modular and semi-simple aspects (cf. [1] and [6]). Thus the property of subgroups being balanced in modular group rings is crucial for the investigation of nice composition series and nice bases in such rings (see, for instance, [9] or [5]). Moreover, the balanced subgroups play an important role for the quasi-completeness (e. g. [2, 3]) and torsion-completeness (e. g. [4]) in group algebras by using either an algebraical or topological technique in terms of bounded convergent Cauchy sequences. The query for the balanced property of S(KH) in S(KG) when KG is semisimple, such that G is p-primary and K is either a field having arbitrary characteristic or is a special ring of zero characteristic, is considered and settled in some way by us in [4]. In [9] and [8], May and Hill-Ullery studied the case when R is a field, whereas we here investigate the general situation which cannot be treated by similar reasons.