Ordinals in topological groups

We show that if an uncountable regular cardinal tau and tau + 1 embed in a topological group G as closed subspaces then G is not normal. We also prove that an uncountable regular cardinal cannot be embedded in a torsion free Abelian group that is hereditarily normal. These results are corollaries to our main results about ordinals in topological groups. To state the main results, let tau be an uncountable regular cardinal and G a T-1 topological group. We prove, among others, the following statements: (1) If tau and tau+1 embed closedly in G then tau x (tau+1) embeds closedly in G; (2) If tau embeds in G, G is Abelian, and the order of every non-neutral element of G is greater than 2(N) - 1 then Pi(i is an element of N) tau embeds in G; (3) The previous statement holds if tau is replaced by tau + 1; (4) If G is Abelian, algebraically generated by tau + 1 subset of G, and the order of every element does not exceed 2(N) - 1 then Pi(i is an element of N)(tau + 1) is not embeddable in G.