A group under MAcountable whose square is countably compact but whose cube is not

We show under MA(countable) the existence of a countable subgroup E of 2(c) such that the group H generated by E and G = (x is an element of 2(c): supp x is bounded in c) is a group as in the title. We also show under MA( countable) that for each k is an element of N there exists a countable family (E-n: n is an element of N) of countable subgroups of 2(c) such that if H-n = E-n + G, then for each subset F of N of size I;, Pi(n is an element of F) H-n is countably compact. while for each subset F of N of size k + 1, Pi(n is an element of F) H-n is not countably compact. (C) 1999 Elsevier Science B.V. All rights reserved.