Quasi-inverse endomorphisms

Abstract

Abstract. Greither and Pareigis have established a connection between Hopf Galois structures on a Galois extension L=K with Galois group G, and the regular subgroups of the group of permutations on G, which are normalized by G. Byott has rephrased this connection in terms of certain equivalence classes of injective morphisms of G into the holomorphs of the groups N with the same cardinality of G.Childs and Corradino have used this theory to construct such Hopf Galois structures, starting from fixed-point-free endomorphisms of G that have abelian images. In this paper we show that a fixed-point-free endomorphism has an abelian image if and only if there is another endomorphism that is its inverse with respect to the circle operation in the near-ring of maps on G, and give a fairly explicit recipe for constructing all such endomorphisms.

Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures

“…Note that an abelian map is constant on conjugacy classes. A classification of fixed point free abelian maps on certain classes of finite groups is well understood: see [3,7,14].…”

“…We first claim that ψ is a fixed point free abelian endomorphism. (Indeed, it is the quasi-inverse of Ψ as described in [7] and [3].) Note that gΨ(g −1 )hΨ(g) Ψ gΨ(g −1 )hΨ(g)…”

Computation of orders and cycle lengths of automorphisms of finite solvable groups

“…Our Algorithm 2 solves the first problem in the special case where S is a (finite) solvable group G and f is a bijective affine map of G. Without giving a detailed analysis, we note that the other two problems for (S, f ) = (G, ϕ), a finite solvable group together with an endomorphism, admit deterministic solution algorithms with complexity polynomial in Coll(P ), P the refined consistent polycyclic presentation through which G is given. This is because by [6,Theorem 4.2] and Lagrange's theorem, the preperiod length of any g ∈ G under ϕ is at most ⌊log 2 |G|⌋, and in particular, the subgroup of G consisting of the periodic points of ϕ is just the image of ϕ ⌊log 2 |G|⌋ .…”

On the dynamics of endomorphisms of finite groups

“…Proof. For (1) and (2), note that it suffices to show that nil(ϕ) and per(ϕ) are subgroups (and nil(ϕ) normal), which is clear by observing that nil(ϕ) is the maximum (with respect to inclusion) of the ascending chain of normal subgroups (ker (m) (ϕ)) m∈N and per(ϕ) is the minimum of the descending chain of subgroups (im(ϕ n )) n∈N . From these observations, (3) immediately follows from the group version of Fitting's Lemma stated and proved as Theorem 4.2 in [2], and (4) is clear by (3) and the structure of semidirect products.…”