Some properties of central kernel quotient of a group

The authors, in 2016, introduced the notion of central autocommutator subgroup of a group [Formula: see text] and proved some new results concerning the central kernel subgroup of [Formula: see text], which was introduced earlier by Haimo in [Formula: see text]. In this paper, we establish some results on central kernel quotient of [Formula: see text]. Finally, we introduce the concept of central autonilpotent (henceforth [Formula: see text]-nilpotent) groups and determine its relationship with the nilpotency of the subgroups of central automorphism of [Formula: see text].

Some finite p-groups with central automorphism group of minimal order

Let [Formula: see text] be a finite [Formula: see text]-group and let [Formula: see text] be the set of all central automorphisms of [Formula: see text] For any group [Formula: see text], the center of the inner automorphism group, [Formula: see text], is always contained in [Formula: see text] In this paper, we study finite [Formula: see text]-groups [Formula: see text] for which [Formula: see text] is of minimal possible, that is [Formula: see text] We characterize the groups in some special cases, including [Formula: see text]-groups [Formula: see text] with [Formula: see text], [Formula: see text]-groups with an abelian maximal subgroup, metacyclic [Formula: see text]-groups with [Formula: see text], [Formula: see text]-groups of order [Formula: see text] and exponent [Formula: see text] and Camina [Formula: see text]-groups.

Some finite p-groups with central automorphism group of non-minimal order

We characterize finite [Formula: see text]-groups of order up to [Formula: see text] whose central automorphisms fixing the center element-wise are all inner.

Central automorphisms of free nilpotent Lie algebras

Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

On the Nilpotency and Solubility of the Central Automorphism Group of a Finite Group

In this paper we completely study the nilpotency and solubility of the group Aut c(G) of central automorphisms of a finite group G.

IRREDUCIBLE COXETER GROUPS

We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that an indefinite irreducible Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. Let W be a Coxeter group. Write W = WX1 × ⋯ × WXb × WZ3, where WX1, … , WXb are non-spherical irreducible Coxeter groups and WZ3 is a finite one. By a classical result, known as the Krull–Remak–Schmidt theorem, the group WZ3 has a decomposition WZ3 = H1 × ⋯ × Hq as a direct product of indecomposable groups, which is unique up to a central automorphism and a permutation of the factors. Now, W = WX1 × ⋯ × WXb × H1 × ⋯ × Hq is a decomposition of W as a direct product of indecomposable subgroups. We prove that such a decomposition is unique up to a central automorphism and a permutation of the factors. Write W = WX1 × ⋯ × WXa × WZ2 × WZ3, where WX1, … , WXa are indefinite irreducible Coxeter groups, WZ2 is an affine Coxeter group whose irreducible components are all infinite, and WZ3 is a finite Coxeter group. The group WZ2 contains a finite index subgroup R isomorphic to ℤd, where d = |Z2| - b + a and b - a is the number of irreducible components of WZ2. Choose d copies R1, … , Rd of ℤ such that R = R1 × ⋯ × Rd. Then G = WX1 × ⋯ × WXa × R1 × ⋯ × Rd is a virtual decomposition of W as a direct product of strongly indecomposable subgroups. We prove that such a virtual decomposition is unique up to commensurability and a permutation of the factors.