Indecomposable Linear Groups

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 obtained by L. Nazarova [4,5]. This description was significantly simplified and clarified after the discovery of the connection between the Z-representations of the group G and the representations of a certain graph with five vertices defined on linear spaces over the field of two elements [2]. Let K = Z, Z (2) , or Z 2 be the ring of rational integers, the localization of Z at the prime 2 and the ring of 2-adic integers, respectively. Let G = a, b ∼ = C 2 × C 2 be a noncyclic group of order 4. Let M n (n > 1) be the set of all polynomials f (x) of degree n over the field of two elements F 2 which satisfy one of the following conditions: • f (x) is irreducible over F 2 ; • f (x) is a power of a nonlinear irreducible polynomial over F 2. Define an action of the group Aut(G) = σ 1 , σ 2 |σ 1 (a) = b, σ 1 (b) = a, σ 2 (a) = a, σ 2 (b) = ab, a, b ∈ G ∼ = S 3