Inverse Limits of Solvable Groups

Inverse Limits of Solvable Groups

In this paper we generalize to groups of Galois type some results of P. Hall on finite solvable groups [l; 2; 3]. We need, in a modified form, some results of van Dantzig: the definition of supernatural numbers (which are related to van Dantzig's universal numbers) and Theorem 5, which he proved for ordinary ¿>-Sylow subgroups [6]. Lemmas 1 and 4 and the method of proof in Theorem 5 are due to ...

Inverse Limits and Profinite Groups

Standard examples of inverse limits arise from sequences of groups, with maps between them: for instance, if we have the sequence Gn = Z/pZ for n ≥ 0, with the natural quotient maps πn+1 : Gn+1 → Gn, the inverse limit consists of tuples (g0, g1, . . . ) ∈ ∏ n≥0Gn such that πn+1(gn+1) = gn for all n ≥ 0. This is a description of the p-adic integers Zp. It is clear that more generally if the Gn a...

On cardinalities in quotients of inverse limits of groups

Let λ be א0 or a strong limit of cofinality א0. Suppose that 〈Gm, πm,n : m ≤ n < ω〉 and 〈Hm, π t m,n : m ≤ n < ω〉 are projective systems of groups of cardinality less than λ and suppose that for every n < ω there is a homorphism σ : Hn → Gn such that all the diagrams commute. If for every μ < λ there exists 〈fi ∈ Gω : i < μ〉 such that i 6= j =⇒ fif −1 j 6∈ σω(Hω) then there exists 〈fi ∈ Gω : i ...

Classification of solvable groups with a given property

In this paper we classify all finite solvable groups satisfying the following property P5: their orders of representatives are set-wise relatively prime for any 5 distinct non-central conjugacy classes.

Generalized Covering Groups and Direct Limits