Counting compact group topologies

Given a group K  , the symbol cgt(K)cgt(K) denotes the set of Hausdorff compact group topologies on K  . The authors ask: when |K|=κ≥ω|K|=κ≥ω, what are the possible cardinalities of a pairwise homeomorphic subset [resp., a pairwise nonhomeomorphic subset] of cgt(K)cgt(K)? Revisiting (sometimes, improving) theorems of Halmos, Hulanicki, Fuchs, Hawley, Chuan/Liu, Kirku, and Shtern, the authors show inter alia:1. |cgt(K)|≤2κ|cgt(K)|≤2κ, and every pairwise nonhomeomorphic V⊆cgt(K)V⊆cgt(K) satisfies |V|≤κ|V|≤κ.2. If K   is infinite, abelian and divisible with cgt(K)≠∅cgt(K)≠∅, there is a pairwise homeomorphic V⊆cgt(K)V⊆cgt(K) such that |V|=|cgt(K)|=2κ|V|=|cgt(K)|=2κ. In particular for λ≥ωλ≥ω and K=RλK=Rλ or K=TλK=Tλ, there is a pairwise homeomorphic V⊆cgt(K)V⊆cgt(K) such that |V|=2|K|=2(2λ)|V|=2|K|=2(2λ).3. [K   not necessarily abelian] If some T∈cgt(K)T∈cgt(K) is connected and the connected component Z0(K,T)Z0(K,T) of the center of (K,T)(K,T) satisfies π1(Z0(K,T))≠{0}π1(Z0(K,T))≠{0}, then |cgt(K)|=2|K||cgt(K)|=2|K|.4. Corollary to 3: Every nonsemisimple compact connected Lie group (K,T)(K,T) satisfies |cgt(K)|=2c|cgt(K)|=2c.5. [Hulanicki] In ZFC: there is a pairwise nonhomeomorphic V⊆cgt(T)V⊆cgt(T) such that |V|=c|V|=c.6. Concerning pairwise nonhomeomorphic V⊆cgt(R)V⊆cgt(R): |V|=ω|V|=ω occurs in ZFC; |V|=ω|V|=ω is best possible in ZFC + CH; and |V|>ω|V|>ω is consistent with ZFC.7. A compact abelian group (K,T)(K,T) satisfies cgt(K)={T}cgt(K)={T} if and only if each automorphism of K   is a TT-homeomorphism.