Laffey Automorphisms Of Finite Groups

The automorphism groups of finite groups are one of the important and difficult research subjects in group theory. In recent years, it has become increasingly active. Let G be a group and letα∈Aut(G). If g^αg = gg^αfor all g∈G, then we calla is a commuting automorphism of G. From the definition it is not difficult to see that commuting automorphisms are a kind of generation of central automorphisms. The set of all the commuting automorphisms of G is denoted by A(G). It is not the subgroup of Aut(G) in general.In 2002 in their paper[16] Deaconescu etc. have deeply investigated into commuting automorphisms and obtained a series of important results. A key technique appeared in the article is the so-called the Laffey lemma, i.e., the automorphism satisfies the condition that [x^α,y] = [x, y^α],(?) x,y∈G,α∈L(G). This thesis calls theautomorphism which satisfies the above formula as a Laffey automorphism of G, and denotes by L(G) the set of all Laffey automorphisms. Therefore, we may extend the most results of commuting automorphisms to Laffey automorphisms.The following theorems are the main results of this paper:We first give a criterion for an automorphism to be a Laffey automorphism:Theorem 1. Let G be a group and letα∈Aut(G). Ifα²âˆˆAut_c(G), thenα∈L(G) if and only if G'≤C_G(α).From the perspective of generators we obtain some conditions of Laffey automorphisms:Theorem 2. Let G be a group such that G = and letα∈Aut(G), thenα∈L(G) if and only if [x^α, y] = [x, y^α], (?) x, y∈X.The next result shows the relationship between Laffey automorphisms and commuting automorphisms. It gives conditions for Laffey automorphisms to be commuting automorphisms:Theorem 3. Let G be a group. Then:(1) If the |G'| is odd, then L(G) = A(G) and [G,α]≤E₂(G),(?)α∈L(G).(2) Ifα∈L(G), thenα²âˆˆA(G).In [16] it has given many properties of commuting automorphisms. The most important conclusions can be used to generalize Laffey automorphisms. This is the following theorem: Theorem 4. Let G be a group and letα,β∈L(G). Then:(1) L(G) is closed under powers, that is,αⁿ∈L(G), for all n∈N;(2) L(G) is closed under conjugates, that is, ifγ∈Aut(G), then (γ)^-1αγ∈L(G);(3) (αβ)ⁿα,β(αβ)ⁿ∈L(G) for all n≥0;(4) [α,β] fixes all elements of G';(5)αβ∈L(G) implies that [α,β]∈Aut_c(G);(6) L(G) is a union of cosets modulo Cent(G) ;(7) Ifα∈L(G), thenα²âˆˆCent(G);(8)α²âˆˆAut_c(G) if and only if G'≤C_G(α).