Denote by G the class of all Abelian Hausdorff topological groups. A group G is-an-element-of G is minimal (totally minimal) if every continuous group isomorphism (homomorphism) i: G --> H of G onto H is-an-element-of G is open. For G is-an-element-of G let K(G) be the smallest cardinal tau greater-than-or-equal-to 1 such that the minimality of G(tau) implies the minimality of all powers of G. For Q subset-of G, Q not-equal phi, we set kappa(Q) = sup{kappa(G): G is-an-element-of G} and denote by alpha(Q) the smallest cardinal tau greater-than-or-equal-to 1 having the following property: If {G(i): i is-an-element-of I} subset-of Q, I not-equal phi, and each subproduct PI{G(i): i is-an-element-of J}, with J subset-of 1, J not-equal phi, and Absolute value of J less-than-or-equal-to tau, is minimal, then the whole product PI{G(i): i is-an-element-of I} is minimal. These definitions are correct, and kappa(G) less-than-or-equal-to 2omega and kappa(Q) less-than-or-equal-to alpha(Q) less-than-or-equal-to 2omega for all G is-an-element-of G and any Q subset-of G, Q not-equal phi, while it can happen that kappa(Q) < alpha(Q) for some Q subset-of G. Let C = {G is-an-element-of G : G is countably compact{ and P = {G is-an-element-of G: G is pseudocompact}. If G is-an-element-of C is minimal, then G x H is minimal for each minimal (not necessarily Abelian) group H ; in particular, G(n) is minimal for every natural number n . We show that alpha(C) = omega, and so either kappa(C) = 1 or kappa(C) = omega. Under Lusin's Hypothesis 2omega1 = 2omega we construct {G(n): n is-an-element-of N} subset-of P and H is-an-element-of P such that: (i) whenever n is-an-element-of N, G(n)n is totally minimal, but G(n)n+1 is not even minimal, so kappa(G(n)) = n+1 ; and (ii) H(n) is totally minimal for each natural number n , but H(omega) is not even minimal, so kappa(H) = omega. Under MA + -CH, conjunction of Martin's Axiom with the negation of the Continuum Hypothesis, we construct G is-an-element-of P such that G(tau) is totally minimal for each T < 2omega, while G2omega is not Minimal, so kappa(G) = 2omega. This yields alpha(P) = kappa(P) = 2omega under MA + -CH. We also present an example of a noncompact minimal group G is-an-element-of C, which should be compared with the following result obtained by the authors quite recently: Totally minimal groups G is-an-element-of C are compact.