[Paper Review] Abelian Groups, Homomorphisms and Central Automorphisms of Nilpotent Groups

Ayan Mahalanobis|ArXiv.org|Feb 14, 2006

Coding theory and cryptography14 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for the group of central automorphisms of a finite purely non-abelian $p$-group to be abelian, by reducing the problem to studying c-maps—homomorphisms between abelian groups satisfying a commutativity condition. The key result characterizes such groups via the image of a specific subgroup $\mathcal{R}$ under the map $\lambda: Z(G) \to G/G'$, showing that $\lambda(\mathcal{R}) = (G/G')^{p^k}$ for $c \leq k < n_1$ and $\mathcal{R}/(Z(G) \cap G')$ cyclic.

ABSTRACT

In this paper we find a necessary and sufficient condition for a finite nilpotent group to have an abelian central automorphism group.

Motivation & Objective

• To determine a necessary and sufficient condition for the group of central automorphisms of a finite purely non-abelian $p$-group to be abelian.
• To extend Adney and Yen's result on $p$-groups of class 2 to arbitrary finite purely non-abelian $p$-groups.
• To reduce the non-abelian automorphism problem to a problem in abelian group theory via homomorphisms between $G/G'$ and $Z(G)$.
• To characterize c-maps—homomorphisms $\lambda: A \to B$ such that $f\lambda g = g\lambda f$ for all $f,g \in \text{Hom}(B,A)$—and apply them to central automorphisms.
• To establish a structural link between the commutativity of central automorphisms and the image of the subgroup $\mathcal{R}$ under the map $\lambda$.

Proposed method

• Use the one-to-one correspondence between central automorphisms $\sigma \in \text{Aut}_c(G)$ and homomorphisms $\phi_\sigma: G \to Z(G)$, which factor through $G/G'$.
• Define the map $\lambda: Z(G) \to G/G'$ by $\lambda(x) = xG'$, which links the center and the abelianization of $G$.
• Apply Theorem 2.1 to show that $\text{Aut}_c(G)$ is abelian if and only if $\lambda$ is a c-map, i.e., $f\lambda g = g\lambda f$ for all $f,g \in \text{Hom}(G/G', Z(G))$.
• Reduce the problem to finite abelian $p$-groups $A = Z(G)$ and $B = G/G'$, decomposed into cyclic $p$-groups.
• Define $\mathcal{R} \subseteq A$ as the set of elements of order $p^{n_1}$, where $n_1 = \text{exp}(A)$, and analyze $\lambda(\mathcal{R})$.
• Prove that $\lambda$ is a non-trivial c-map if and only if $\lambda(\mathcal{R}) = B^{p^k}$ for $c \leq k < n_1$ and $\mathcal{R}/\ker(\lambda)$ is cyclic, where $c = \text{exp}(Z(G) \cap G')$.

Experimental results

Research questions

• RQ1 When is the group of central automorphisms of a finite purely non-abelian $p$-group abelian?
• RQ2 What structural conditions on the center $Z(G)$ and the abelianization $G/G'$ ensure that the induced map $\lambda: Z(G) \to G/G'$ is a c-map?
• RQ3 How does the image of the subgroup $\mathcal{R}$ of maximal order elements in $Z(G)$ relate to the commutativity of $\text{Aut}_c(G)$?
• RQ4 In what way does the condition $\lambda(\mathcal{R}) = (G/G')^{p^k}$ for $c \leq k < n_1$ and $\mathcal{R}/(Z(G) \cap G')$ cyclic characterize abelian central automorphism groups?
• RQ5 How does this characterization generalize Adney and Yen’s result for $p$-groups of class 2?

Key findings

• The group of central automorphisms of a finite purely non-abelian $p$-group $G$ is abelian if and only if $\lambda(\mathcal{R}) \subseteq (G/G')^{p^{n_1}}$ or $\lambda(\mathcal{R}) = (G/G')^{p^k}$ for $c \leq k < n_1$ with $\mathcal{R}/(Z(G) \cap G')$ cyclic.
• The condition $\lambda(\mathcal{R}) = (G/G')^{p^k}$ with $c \leq k < n_1$ generalizes Adney and Yen’s result for $p$-groups of class 2, where $\lambda(\mathcal{R}) = (G/G')^{p^c}$ and $\mathcal{R}/G' \cong \langle x_1^{p^c}G' \rangle$.
• A non-trivial c-map $\lambda: A \to B$ exists if and only if $\lambda(\mathcal{R}) = p^k B$ with $c \leq k < n_1$ and $\mathcal{R}/\ker(\lambda)$ cyclic, where $A = Z(G)$, $B = G/G'$, and $c = \text{exp}(Z(G) \cap G')$.
• If $\text{exp}(\ker(\lambda)) = \text{exp}(\text{coker}(\lambda))$, then $\lambda(\mathcal{R}) = p^c B$, which recovers the class 2 case.
• The subgroup $\mathcal{R} \subseteq Z(G)$ consists of all elements of order $p^{n_1}$, where $n_1 = \text{exp}(Z(G))$, and plays a central role in determining the commutativity of $\text{Aut}_c(G)$.
• The proof relies on analyzing the action of homomorphisms $f,g \in \text{Hom}(B,A)$ on $\lambda$, showing that $f\lambda g = g\lambda f$ holds precisely when $\lambda(\mathcal{R})$ lies in a specific power of $B$ and the quotient $\mathcal{R}/\ker(\lambda)$ is cyclic.