Given a graph G = (V, E), with each subset X of V is associated the subgraph G(X) of G induced by X. A subset I of V is an interval of G provided that for any a, b ∈ I and x ∈ V \ I , {a, x} ∈ E if and only if {b, x} ∈ E. For example, ∅, {x}, where x ∈ V , and V are intervals of G called trivial intervals. A graph is indecomposable if all its intervals are trivial; otherwise, it is decomposable...

Abstract. A permutation a1a2 . . . an is indecomposable if there does not exist p < n such that a1a2 . . . ap is a permutation of {1, 2, . . . , p}. We consider the probability that a permutation of Sn with m cycles is indecomposable and prove that this probability is monotone non-increasing in n. We compute also the asymptotic probability when n goes to infinity with m/n tending to a fixed rat...

Let G be a noncyclic group of order 4, and let K = Z, Z (2) and Z 2 be the ring of rational integers, the localization of Z at the prime 2 and the ring of 2-adic integers, respectively. We describe, up to conjugacy, all of the indecomposable subgroups in the group GL(m, K) which are isomorphic to G. The first explicit description of the Z-representations of the noncyclic group G of order 4 was ...