A bound for Hall's criterion for nilpotence in semi-abelian categories

TL;DR

This paper improves the bound in Hall's criterion for nilpotence within semi-abelian categories, establishing the optimal bound and extending Gray's main theorem in this mathematical context.

Contribution

It provides the best possible bound for Hall's criterion for nilpotence in semi-abelian categories, enhancing Gray's theorem.

Findings

Improved the bound in Hall's criterion for nilpotence

Proved the bound is optimal

Extended Gray's main theorem

Abstract

In this paper, we focus on Hall's criterion for nilpotence in semi-abelian categories, and we improve the bound of Gray's main theorem of [3,Theorem 3.4] (see Main Theorem). And this bound is best possible.

Equations

$$a\le b⟺a\vee b=b$$

$$g^i(x)\cdot g^j(x)\le g^{i+j+1}(x)$$

$$g^i(x)\cdot g^j(x)$$

$$f^n(a\cdot b)\le \underset{i=0}{\overset{n}{\bigvee }}{f}^i(a)\cdot f^{n-i}(b).$$

$$f(a\cdot b)\le (f(a)\cdot b)\vee (a\cdot f(b))$$

$$f^{n-1}(a\cdot b)\le \underset{i=0}{\overset{n-1}{\bigvee }}{f}^i(a)\cdot f^{n-1-i}(b).$$

$$f^n(a\cdot b)=f(f^{n-1}(a\cdot b))\le f(\underset{i=0}{\overset{n-1}{\bigvee }}{f}^i(a)\cdot f^{n-1-i}(b))\le \underset{i=0}{\overset{n-1}{\bigvee }}f(f^i(a)\cdot f^{n-1-i}(b)).$$

$$f^n(a\cdot b)$$

$$f^{t_k}(y)\le g^k(x)$$

$$f^{t_r}(y)\le g^r(x)$$

$$f^{t_k}(y)$$

$$2(j-1)m-m-(j-1)+1\le i<2jm-m-j+1.$$

$$f^i(x)\le f^i(y)\le f^{2(j-1)m-m-(j-1)+1}(y).$$

$$f^{2(j-1)m-m-(j-1)+1}(y)=f^{t_{j-1}}(y)\le g^{j-1}(x).$$

$$f^{t_k-m-i}(x)\le f^{t_k-m-i}(y).$$

$$t_k-m-i$$

$$f^{t_k-m-i}(x)\le f^{t_k-m-i}(y)$$

$$f^i(x)\cdot f^{t_k-m-i}(x)\le g^{j-1}(x)\cdot g^{k-j}(x)\le g^{k-j+j-1+1}(x)=g^k(x).$$

$$f^{t_k}(y)$$

$$\begin{array}{ccccc}A& \stackrel{⟨1,0⟩}{\to }& A\times B& \stackrel{⟨0,1⟩}{\to }& B\\ f\downarrow & & \phi \downarrow & & g\downarrow \\ C& \stackrel{=}{\to }& C& \stackrel{=}{\to }& C\end{array}$$

$$f^{cd+(c-1)(d-1)}(E)\le g^c(N).$$

Full text

Key Words: Hall’s criterion for nilpotence; Semi-abelian. 2000 Mathematics Subject Classification:   $\cdot$   $\cdot$

1. Introduction

In [3], Gray has proved a wide generalization of P. Hall’s theorem about nilpotent groups: a group $G$ is nilpotent if it has a normal subgroup $N$ such that $G/[N,N]$ and $N$ is nilpotent. Gray’s generalization is in a semi-abelian category [8] which satisfies some properties[3, Section 3]. Moreover, Gray’s main theorem gives a bound of the nilpotency class about the similar objects in algebraically coherent semi-abelian category (see [3, Theorem 3.4]). In this note, we improve the bound as follows.

Main Theorem**.**

Let $\mathbb{C}$ be an algebraically coherent semi-abelian category and let $p:E\to B$ be an extension of a nilpotent object $B$ in $\mathbb{C}$. If the kernel of $p$ is contained in the Huq commutator $[N,N]_{N}$ of a nilpotent normal subobject $N$ of $E$, and if $N$ is of nilpotency class $c$ and $B$ is of nilpotency class $d$, then $E$ is of nilpotency class at most $cd+(c-1)(d-1)$.

Here, the definition of algebraically coherent semi-abelian category can be found in [3]. Examples of algebraically coherent semi-abelian categories include the categories of groups, rings, Lie algebra over a commutative ring, and others categories in [9]. And the bound in the categories of groups is found by [11, Theorem 1.].

$Structure~{}of~{}the~{}paper:$ After recalling the basic definitions and properties of commutator semi-lattices in Section 2, and we introduce semi-abelian categories and commutators in Section 3. In Section 4, we prove Main Theorem.

2. Jacobi commutator semi-lattices

In this section we collect some known results about commutator semi-lattices. For the background theory of commutator semi-lattices, we refer to [3].

First, let us begin with semi-lattices.

Definition 2.1**.**

A $\mathrm{semi}$-$\mathrm{lattice}$ is a triple $(X,\leq,\vee)$ where $(X,\leq)$ is a poset, and $\vee$ is a binary operation on $X$ satisfying:

(a) for each $a\in X$, $a\vee a=a$;

(c) for each $a,b\in X,a\vee b=b\vee a$;

(d) for each $a,b,c\in X,(a\vee b)\vee c=a\vee(b\vee c)$.

Moreover, a semi-lattice $(X,\leq,\vee)$ is called $\mathrm{join}$ if

[TABLE]

for each $a,b\in X$.

Definition 2.2**.**

A $\mathrm{commutator~{}semi}$-$\mathrm{lattice}$ is a triple $(X,\leq,\cdot)$ where $X$ is a set, $\leq$ is a binary relation on $X$, and $\cdot$ is a binary operation on $X$ satisfying:

(a) $(X,\leq)$ is a join semi-lattice;

(b) the operation $\cdot$ is commutative;

(c) for each $a,b\in X,a\cdot b\leq b$;

(d) for each $a,b,c\in X,a\cdot(b\vee c)=(a\cdot b)\vee(a\cdot c)$.

Remark 2.3**.**

Let $(X,\leq,\cdot)$ be a commutator semi-lattice, and $x\in X$. Then the map $x\cdot-:X\to X$ defined by $y\mapsto x\cdot y$ is order preserving.

Proof.

Let $y\leq z\in X$, then $y\vee z=z$. Since $(x\cdot y)\vee(x\cdot z)=x\cdot(y\vee z)$ by above definition (d), we have $(x\cdot y)\vee(x\cdot z)=x\cdot(y\vee z)=x\cdot z$. Hence, $x\cdot y\leq x\cdot z$. ∎

Definition 2.4**.**

A commutator semi-lattice $(X,\leq,\cdot)$ is a $\mathrm{Jacobi}$ commutator semi-lattice if

(a) for each $a,b,c\in X,a\cdot(b\cdot c)\leq((a\cdot b)\cdot c)\vee(b\cdot(a\cdot c))$;

Example 2.5**.**

Let $G$ be a group, and let $X$ be the set of all normal subgroups of $G$. For each $M,N\in X$, we can define that $N\cdot M=[M,N]$ and $N\vee M=NM$. It is easy to see that $(X,\leq,\cdot)$ is a Jacobi commutator semi-lattice.

Definition 2.6**.**

A derivation of a commutator semi-lattice $(X,\leq,\cdot)$ is a map $f:X\to X$ which preserves joins and satisfies:

(a) for each $a,b\in X,f(a\cdot b)\leq(f(a)\cdot b)\vee(a\cdot f(b))$.

A derivation $f$ of a commutator semi-lattice $(X,\leq,\cdot)$ is an inner derivation if there exists $x$ in $X$ such that $f=x\cdot-$, that is, for each $a$ in $X,f(a)=x\cdot a$.

Remark 2.7**.**

Let $f$ be a derivation of commutator semi-lattice $(X,\leq,\cdot)$. For each $a,b$ in $X$ and $a\leq b$, then $f(a)\leq f(b)$.

Proof.

Since $a\leq b$, we have $a\vee b=b$. Also $f$ is a derivation, thus $f$ preserves joins. Hence, $f(b)=f(a\vee b)=f(a)\vee f(b)$. So $f(a)\leq f(b)$. ∎

Proposition 2.8**.**

Let $g$ be the inner derivation of a Jacobi commutator semi-lattice $(X,\leq,\cdot)$, let $x$ be an elements of $X$, and let $g$ be defined for each $s$ in $X$ by $g(s)=x\cdot s$. Then

[TABLE]

for each each non-negative integers $i$ and $j$.

Proof.

The proof is by induction on $j$. For $j=0$, we can see $g^{i}(x)\cdot g^{0}(x)=g^{i}(x)\cdot x=x\cdot g^{i}(x)=g^{i+1}(x)$.

We can see that

[TABLE]

Hence, we get the proof. ∎

Proposition 2.9**.**

Let $f$ be a derivation of commutator semi-lattice $(X,\leq,\cdot)$. For each $a,b$ in $X$ and for each non-negative integer $n$, we have

[TABLE]

Proof.

The proof is by induction on $n$. For $n=0$, it follows by $f^{0}(a\cdot b)=a\cdot b\leq a\cdot b=f^{0}(a)\cdot f^{0}(b)$. For $n=1$, we can see that

[TABLE]

for each $a,b\in X$ by the definition of derivation.

Now, we can assume that the proposition hold for $n-1$. That means

[TABLE]

By Remark 2.5 and the definition of derivation, we have

[TABLE]

So, we can see that

[TABLE]

Hence, we get the proof. ∎

Proposition 2.10**.**

Let $f$ be a derivation of a Jacobi commutator semi-lattice $(X,\leq,\cdot)$ bounded above by $1_{X}$, let $x$ and $y$ be an elements of $X$, and let $g$ be the inner derivation of $(X,\leq,\cdot)$ defined for each $s$ in $X$ by $g(s)=x\cdot s$. If $x\leq y$ and for some positive integer $m$, $f^{m}(y)\leq g(x)$, then for each positive integer $k$,

[TABLE]

where $t_{k}=km+(k-1)(m-1).$

Proof.

The proof is by induction on $k$.

Step 1. For $k=1$, we can see $t_{1}=m$. So we can see the condition $f^{t_{k}}(y)\leq g^{k}(x)$ makes this case holds.

Step 2. If $k>1$, then for $r\leq k-1$, we can assume that the proposition hold when $r\leq k-1$. That means that

[TABLE]

for each $1\leq r\leq k-1$.

Step 3. For $k$, we can see that $f^{t_{k}}(y)\leq f^{t_{k}-m}(f^{m}(y))\leq f^{t_{k}-m}(g(x))$ by Remark 2.7. And by Proposition 2.9, we have

[TABLE]

Now, we will consider $f^{i}(x)\cdot f^{t_{k}-m-i}(x)$ for each $i$. For each $i$, there exists $1\leq j\leq k$ such that

[TABLE]

For $f^{i}(x)$, we can see that

[TABLE]

Here, $2(j-1)m-m-(j-1)+1=t_{j-1}$. But $j-1\leq k-1$, thus

[TABLE]

For $f^{t_{k}-m-i}(x)$, we can see that

[TABLE]

But

[TABLE]

Since $i<2jm-m-j+1$, we have $jm-m-j-i\geq 0$. Hence, we have

[TABLE]

Hence, we have

[TABLE]

Therefore, we can see

[TABLE]

and we prove this proposition. ∎

The above proof follows [11, Theorem 1].

3. Semi-abelian categories

In this section we collect some known results about semi-abelian categories. For the background theory of semi-abelian categories, we refer to [8].

We introduce the Huq commutator as follows, and we use the notations in [3].

Definition 3.1**.**

Let $\mathcal{C}$ be a semi-abelian category. Denote by [math] a zero object in $\mathcal{(}C)$, and denote by [math] each zero morphism, that is, a morphism which factors through a zero object. For each $A,B\in\mathrm{Ob}(\mathcal{C})$, we have a product $A\times B\in\mathrm{Ob}(\mathcal{C})$. By the definition of product, we can write $\langle 1,0\rangle:A\to A\times B$ and $\langle 0,1\rangle:B\to A\times B$ for the unique morphisms with $\pi_{1}\langle 1,0\rangle:=1_{A},\pi_{2}\langle 1,0\rangle:=0,\pi_{1}\langle 0,1\rangle:=0$ and $\pi_{2}\langle 0,1\rangle:=1_{B}$. A pair of morphisms $f:A\to C$ and $g:B\to C$ commute, if there is a morphism $\varphi:A\times B\to C$ making the diagram commute.

[TABLE]

More generally the Huq commutator of $f:A\to C$ and $g:B\to C$ is defined to be the smallest normal subobject $N$ of $C$ such that $qf$ and $qg$ commute, where $q:C\to C/N$ is the cokernel of the associated normal monomorphism $N\to C$.

Theorem 3.2**.**

Let $\mathcal{C}$ be a semi-abelian category. Let $f:A\to C$ and $g:B\to C$ be morphisms in $\mathcal{C}$, then there exists the Huq commutator of $f$ and $g$.

Proof.

See [1] or [3]. ∎

We recall the definition of nilpotent for object in a semi-abelian category $\mathcal{C}$ as follows.

Definition 3.3**.**

For subobjects $S$ and $T$ of an object $C$ in $\mathcal{C}$, we will write $[S,T]_{C}$ for the Huq commutator of the associated monomorphisms $S\to C$ and $T\to C$. Recall also that C is nilpotent if there exists a non-negative integer $n$ such that $\gamma^{n}_{C}(C)=0$, where $\gamma_{C}$ is the map sending $S$ in $\mathbf{Sub}(C)$ to $[C,S]_{C}$ in $\mathbf{Sub}(C)$. The least such $n$ is the nilpotency class of $C$.

4. Proof of the main theorem

In this section, we give a proof of the main theorem.

Theorem 4.1**.**

Let $\mathbb{C}$ be an algebraically coherent semi-abelian category and let $p:E\to B$ be an extension of a nilpotent object $B$ in $\mathbb{C}$. If the kernel of $p$ is contained in the Huq commutator $[N,N]_{N}$ of a nilpotent normal subobject $N$ of $E$, and if $N$ is of nilpotency class $c$ and $B$ is of nilpotency class $d$, then $E$ is of nilpotency class at most $cd+(c-1)(d-1)$.

Proof.

Let $X:=\mathbf{NSub}(E)$(Here, $\mathbf{NSub}(E)$ means all normal subobjects of $E$) and let $f:X\to X$ and $g:X\to X$ be the maps defined by $f(K)=[E,K]_{E}$ and $g(K)=[N,K]_{E}$. Using the proof of [3, Theorem 3.4], we find $f^{d}(E)\leq g(N)$. So by Proposition 2.10, we get

[TABLE]

So, we get the proof because $N$ is of nilpotency class $c$. ∎

Example 4.2**.**

[11, Section 5, Example]** For every pair $c,d$ of positive integers there is a group $G$ of class $cd+(c-1)(d-1)$ which has a normal subgroup $N$ of class $c$ such that $G/[N,N]$ is of class $d$.

ACKNOWLEDGMENTS

The authors are grateful to the website address: ncatlab.org/nlab/show/HomePage for its guidance.