Suppose α is a nonzero cardinal number, I is an ideal on arc connected topological space X, and PαI(X) is the subgroup of pi1(X) (the first fundamental group of X) generated by homotopy classes of αI loops. The main aim of this text is to study PαI(X)s and compare them. Most interest is in α ∈ {ω, c} andI ∈ {P f in(X), {∅}}, where P f in(X) denotes the collection of all finite subsets of X. We denote Pα{∅}(X) with Pα(X). We prove the following statements: • for arc connected topological spaces X and Y if Pα(X) is isomorphic to Pα(Y) for all infinite cardinal number α, then pi1(X) is isomorphic to pi1(Y); • there are arc connected topological spaces X and Y such that pi1(X) is isomorphic to pi1(Y) but Pω(X) is not isomorphic to Pω(Y); • for arc connected topological space X we have Pω(X) ⊆ Pc(X) ⊆ pi1(X); • for Hawaiian earring X, the sets Pω(X), Pc(X), and pi1(X) are pairwise distinct. SoPα(X)s andPαI(X)s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.