Brouwer and cardinalities

This note is a somewhat personal account of a paper that L.E.J. Brouwer published in 1908 and that dealt with the possible cardinalities of subsets of the continuum. That paper is of interest because it represents the first time that Brouwer presented his ideas on foundations in an international forum.I found Brouwer’s notions and arguments at times hard to grasp if not occasionally perplexing. I hope that this note contributes to a further discussion of the definitions and reasonings as presented in Brouwer’s paper.     

Maximal Divisible Subgroups in p-Mixed Modular Abelian Group Rings

The isomorphism structure of the maximal divisible subgroup dVR(G) of the normed unit group VR(G) in a commutative group ring R(G) is completely described only in terms of R and G whenever R is a commutative unital ring of prime characteristic p and G is a p-mixed abelian group. This somewhat extends a result due to Nachev [13 Nachev , N. A. ( 1995 ). Invariants of the Sylow p-subgroup of the unit group of a commutative group ring of characteristic p. Commun. Algebra 23 : 2469 – 2489 . [Google Scholar]] as well as results due to the author [1 Danchev , P. V. ( 2004 ). Maximal divisible subgroups in modular group algebras of p-mixed and p-splitting abelian groups . Rad. Mat. 13 : 23 – 32 . [Google Scholar]] and Kuneva [11 Kuneva , V. ( 2006 ). Maximal divisible subgroups in modular group algebras of p-mixed abelian groups . Compt. rend. Acad. bulg. Sci. 59 : 899 – 902 . [Google Scholar]].