Commuting automorphisms of finite 2-groups of almost maximal class, II

Let [Formula: see text] be a finite [Formula: see text]-group. In our recent paper, it was shown that in a finite [Formula: see text]-group of almost maximal class, the set of all commuting automorphisms, [Formula: see text] is a subgroup of [Formula: see text]. Also, we proved that the minimum coclass of a non-[Formula: see text], [Formula: see text]-group is equal to 3. Using these results, in this paper, we will take of the task of determining when the group of all commuting automorphisms of all finite [Formula: see text]-groups of almost maximal class are equal to the group of all central automorphisms. This determination is not easy. We will prove they are equal, except only for five ones. We show that the minimum order of a [Formula: see text]-group which it’s group of all commuting automorphisms is not equal to it’s group of all central automorphisms is [Formula: see text]. Also, we prove that if [Formula: see text] is a finite [Formula: see text]-group in which [Formula: see text], then the subgroup of right 2-Engel elements of [Formula: see text], [Formula: see text], coincides with the second term of upper central series 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 properties on $$\mathrm{IA_Z}$$-automorphisms of groups

Let G be a group and $$\mathrm{IA}(G)$$ IA ( G ) denote the group of all automorphisms of G, which induce identity map on the abelianized group $$G_{ab}=G/G'$$ G ab = G / G ′ . Also the group of those $$\mathrm{IA}$$ IA -automorphisms which fix the centre element-wise is denoted by $$\mathrm{IA_Z}(G)$$ IA Z ( G ) . In the present article, among other results and under some condition we prove that the derived subgroups of finite p-groups, for which $$\mathrm{IA_Z}$$ IA Z -automorphisms are the same as central automorphisms, are either cyclic or elementary abelian.

Automorphisms of a generalized extraspecial ℚ-group and its extension

Let [Formula: see text] be a set of primes and [Formula: see text] be a ring consisting of all rational numbers as [Formula: see text], where [Formula: see text] and [Formula: see text] are coprime, [Formula: see text] is a [Formula: see text]-number. The additive group of [Formula: see text] is denoted by [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are two sets of primes and [Formula: see text] is a nonzero integer. Let [Formula: see text] be a generalized extraspecial [Formula: see text]-group as follows: [Formula: see text] Suppose that [Formula: see text] is a direct product of [Formula: see text] and [Formula: see text] copies of [Formula: see text]. Let [Formula: see text] be the normal subgroup of [Formula: see text] consisting of all elements of [Formula: see text] which act trivially on the derived subgroup [Formula: see text] of [Formula: see text], and [Formula: see text] be the normal subgroup of [Formula: see text] consisting of all central automorphisms of [Formula: see text] which also act trivially on the center [Formula: see text] of [Formula: see text]. Then, (i) The extension [Formula: see text] is split; (ii) [Formula: see text]; (iii) If [Formula: see text], then [Formula: see text] and [Formula: see text]; If [Formula: see text], then [Formula: see text] and [Formula: see text].

Equality of derival automorphisms and certain automorphisms in finitely generated groups

Let [Formula: see text] be a group and [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] denote the group of all inner automorphisms, the group of all pointwise inner automorphisms, the group of all central automorphisms and the group of all derival automorphisms of [Formula: see text], respectively. We know that in a finite [Formula: see text]-group [Formula: see text] of class 2, [Formula: see text] if and only if [Formula: see text] is cyclic and [Formula: see text], where [Formula: see text] is the group of all derival automorphisms of [Formula: see text] which fix [Formula: see text] elementwise. In this paper, we characterize all finite nilpotent groups of class 2 for which [Formula: see text] or [Formula: see text] is equal to [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we characterize all finitely generated nilpotent groups of class 2 for which [Formula: see text] is equal to [Formula: see text] and give some interesting corollaries in this regard.

On central automorphisms of crossed modules

A crossed module $(T,G,\partial)$ consist of a group homomorphism $\partial:T\rightarrow G$ together with an action $(g,t)\rightarrow{}^{\,g}t$ of $G$ on $T$ satisfying $\partial(^{\,g}t)=g\partial(t)g^{-1}$ and $\,^{\partial(s)}t=sts^{-1}$, for all $g\in G$ and $s,t\in T$. The term crossed module was introduced by J. H. C. Whitehead in his work on combinatorial homotopy theory. Crossed modules and its applications play very important roles in category theory, homotopy theory, homology and cohomology of groups, algebra, K-theory etc. In this paper, we define Adeny-Yen crossed module map and central automorphisms of crossed modules. If $C^*$ is the set of all central automorphisms of crossed module $(T,G,\partial)$ fixing $Z(T,G,\partial)$ element-wise, then we give a necessary and sufficient condition such that $C^*=I_{nn}(T,G,\partial).$ In this case, we prove $Aut_C(T,G,\partial)\cong Hom((T,G,\partial), Z(T,G,\partial))$. Moreover, when $Aut_C(T,G,\partial)\cong Z(I_{nn}(T,G,\partial)))$, we obtain some results in this respect.

Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras

In the present paper, right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras will be studied. For this purpose, first the structure of the set of all right 2-Engel elements of a Lie algebra will be examined and then, by taking advantage of it, a number of interesting results about central and commuting automorphisms of Lie algebras will be presented. Finally, a characterization of Lie algebras for which the set of central automorphisms is trivial or the set of commuting automorphisms is trivial will be given.