On Free Polyadic Groups

TL;DR

This paper characterizes when a polyadic group derived from a group G, automorphism G, and element b is free, providing a necessary and sufficient condition based on these components.

Contribution

It introduces a precise criterion for the freeness of polyadic groups constructed from given groups, automorphisms, and elements.

Findings

Provides a necessary and sufficient condition for freeness

Links group properties with polyadic group structure

Advances understanding of free polyadic groups

Abstract

In this article, for a polyadic group(G,f),derived from group G by automorphism G and element b, we give a necessary and sufficient condition in terms of the group, the automorphism G, and the element b, in order that the polyadic group becomes free.

Equations

$$f(x_1^{i-1},f(x_i^{n+i-1}),x_{n+i}^{2n-1})=f(x_1^{j-1},f(x_j^{n+j-1}),x_{n+j}^{2n-1})$$

$$f(a_1^{i-1},x,a_{i+1}^n)=b.$$

$$f(\stackrel{(i-1)}{a},x,\stackrel{(n-i)}{a})=x$$

$$f(\stackrel{(n-1)}{x},y)=x.$$

$$f(\stackrel{(i-2)}{x},\overline{x},\stackrel{(n-i)}{x},y)=f(y,\stackrel{(n-i)}{x},\overline{x},\stackrel{(i-2)}{x})=y.$$

$$x\ast y=f(x,\stackrel{(n-2)}{a},y).$$

$$y=f(\overline{a},\stackrel{(n-3)}{x},\overline{x},\overline{a}).$$

$$\{u\widehat{(xu{x}^{-1})}:u\in \mathcal{U},x\in X,u\widehat{(xu{x}^{-1})}\ne 1\}$$

$$s=\frac{k-1}{n-1}+1,$$

$$\{b,{\theta }^j(v_i):1\le i\le s-1,0\le j\le n-2\}$$

$$X=\{u,v_1,\dots ,v_{s-1}\}$$

$$G=F_p^n(X)=\{w\in F(X):\mathrm{ht}(w)\equiv 1(\mathrm{mod}n-1)\},$$

$$F(X)=H\cup Hu\cup \cdots \cup H{u}^{n-2}.$$

$$B=\{u^i\widehat{(x{u}^i{x}^{-1})}\ne 1:0\le i\le n-2,x\in X\}$$

$$u^{i}x\widehat{u^i{x}^{-1}}=u^{i+1}{\widehat{u^{i+1}}}^{-1},$$

$$u^i{v}_j\widehat{u^i{v}_j^{-1}}=u^i{v}_j{u}^{-(i+1)},$$

$$B=\{u^{n-1},u^i{v}_j{u}^{-(i+1)}:0\le i\le n-2,1\le j\le s-1\}.$$

$$f(w_1,\dots ,w_n)=w_1\cdot \cdots \cdot w_n.$$

$$w_1\circ w_2$$

$$w^{-u}$$

$$\theta (w)$$

$$b$$

$$G=Hu=H_u,$$

$$B^{\prime }=\eta (B)=\{u^n,u^i{v}_j{u}^{-i},u^{n-2}{v}_{j}u:0\le i\le n-3,1\le j\le s-1\}$$

$$B^{\prime \prime }=\{u^n,u^i{v}_j{u}^{-i},(u^{n-2}{v}_{j}u)\circ (u^n{)}^{-u}:0\le i\le n-3,1\le j\le s-1\}.$$

$$(u^{n-2}{v}_{j}u)\circ (u^n{)}^{-u}$$

$$B^{\prime \prime }$$

$$(G,f)={\mathrm{der}}_{\theta ,b}(G,\circ )$$

$$w_1\ast w_2=w_1{u}^{-1}{w}_2.$$

$$\alpha (b)=u^n,\alpha ({\theta }^i(v_j))=u^i{v}_j{u}^{-i}.$$

$$\alpha ({\theta }^m(v_{ij}))=u^m\alpha (v_{ij})u^{-m},$$

$$\alpha ({\theta }^m(w))=u^m\alpha (w)u^{-m}.$$

Full text

MSC(2010): Primary 20N15, Secondary 08A99 and 14A99

Keywords: Polyadic groups; $n$-ary groups; Post’s cover; Free polyadic groups; Free group

1. Introduction

In this article, we continue our study of the structure of free polyadic groups and we will answer a question of M. Shahryari. A polyadic group is a natural generalization of the concept of group to the case where the binary operation of group replaced with an $n$-ary associative operation, one variable linear equations in which have unique solutions (see the next section for the detailed definitions). These interesting algebraic objects are introduced by Kasner and Dörnte ([14] and [3]) and studied extensively by Emil Post during the first decades of the last century, [18]. During decades, many articles are published on the structure of polyadic groups. Already homomorphisms and automorphisms of polyadic groups are studied in [15]. A characterization of the simple polyadic groups is obtained by them in [16]. Also representation theory of polyadic groups is studied in [9]. The complex characters of finite polyadic groups are also investigated by the M. Shahryari in [19].

It is known that for every polyadic ($n$-ary) group $(G,f)$, there exists a corresponding ordinary group $(G,\cdot)$, an automorphism $\theta$ of this ordinary group, and an element $b\in G$, the structure of $(G,f)$ in which complectly determined by $(G,\cdot)$, $\theta$, and $b$. M. Shahryari, asked us to find a necessary and sufficient condition for a polyadic group $(G,f)$ to be free in terms of the group $(G,\cdot)$, the automorphism $\theta$, and the element $b$. Our aim is to give an answer to this question.

2. Polyadic groups

This section contains basic notions and properties of polyadic groups as well as some literature. Let $G$ be a non-empty set and $n$ be a positive integer. If $f:G^{n}\to G$ is an $n$-ary operation, then we use the compact notation $f(x_{1}^{n})$ for the elements $f(x_{1},\ldots,x_{n})$. In general, if $x_{i},x_{i+1},\ldots,x_{j}$ is an arbitrary sequence of elements in $G$, then we denote it as $x_{i}^{j}$. In the special case, when all terms of this sequence are equal to a constant $x$, we denote it by $\stackrel{{\scriptstyle(t)}}{{x}}$, where $t$ is the number of terms. We say that an $n$-ary operation is associative, if for any $1\leq i<j\leq n$, the equality

[TABLE]

holds for all $x_{1},\ldots,x_{2n-1}\in G$. An $n$-ary system $(G,f)$ is called an $n$-ary group or a polyadic group, if $f$ is associative and for all $a_{1},\ldots,a_{n},b\in G$ and $1\leq i\leq n$, there exists a unique element $x\in G$ such that

[TABLE]

It is proved that the uniqueness assumption on the solution $x$ can be dropped [5]. Clearly, the case $n=2$ is just the definition of ordinary groups. During this article, we assume that $n>2$. The classical paper of E. Post [18], is one of the first articles published on the subject. In this paper, Post proves his well-known coset theorem. Many basic properties of polyadic groups are studied in this paper. The articles [3] and [14] are among the first materials written on the polyadic groups. Russian reader, can use the book of Galmak [10], for an almost complete description of polyadic groups. The articles [2], [4], [11], and [12] can be used for study of axioms of polyadic groups as well as their varieties.

Note that an $n$-ary system $(G,f)$ of the form $\,f(x_{1}^{n})=x_{1}x_{2}\ldots x_{n}b$, where $(G,\cdot)$ is a group and $b$ a fixed element belonging to the center of $(G,\cdot)$, is an $n$-ary group. Such an $n$-ary group is called $b$-derived from the group $(G,\cdot)$ and it is denoted by $\mathrm{der}_{b}^{n}(G,\cdot)$. In the case when $b$ is the identity of $(G,\cdot)$, we say that such a polyadic group is reduced to the group $(G,\cdot)$ or derived from $(G,\cdot)$ and we use the notation $\mathrm{der}^{n}(G,\cdot)$ for it. For every $n>2$, there are $n$-ary groups which are not derived from any group. An $n$-ary group $(G,f)$ is derived from some group if and only if it contains an element $a$ (called an $n$-ary identity) such that

[TABLE]

holds for all $x\in G$ and for all $i=1,\ldots,n$, see [5].

From the definition of an $n$-ary group $(G,f)$, we can directly see that for every $x\in G$, there exists only one $y\in G$, satisfying the equation

[TABLE]

This element is called skew to $x$ and it is denoted by $\overline{x}$. As Dörnte [3] proved, the following identities hold for all $\,x,y\in G$, $2\leq i\leq n$,

[TABLE]

These identities together with the associativity identities, axiomatize the variety of polyadic groups in the algebraic language $(f,^{-})$.

Suppose $(G,f)$ is a polyadic group and $a\in G$ is a fixed element. Define a binary operation

[TABLE]

Then $(G,\ast)$ is an ordinary group, called the retract of $(G,f)$ over $a$. Such a retract will be denoted by $ret_{a}(G,f)$. All retracts of a polyadic group are isomorphic [8]. The identity of the group $(G,\ast)$ is $\overline{a}$. One can verify that the inverse element to $x$ has the form

[TABLE]

One of the most fundamental theorems of polyadic group is the following, now known as Hosszú -Gloskin’s Theorem. We will use it frequently in this article and the reader can use [6], [7], [13] and [20] for detailed discussions.

Theorem 2.1**.**

Let $(G,f)$ be an $n$-ary group. Then there exists an ordinary group $(G,\cdot)$, an automorphism $\theta$ of $(G,\cdot)$ and an element $b\in G$ such that

1. $\theta(b)=b$,

2. $\theta^{n-1}(x)=bxb^{-1}$, for every $x\in G$,

3. $f(x_{1}^{n})=x_{1}\theta(x_{2})\theta^{2}(x_{3})\cdots\theta^{n-1}(x_{n})b$, for all $x_{1},\ldots,x_{n}\in G$.

According to this theorem, we use the notation $\mathrm{der}_{\theta,b}(G,\cdot)$ for $(G,f)$ and we say that $(G,f)$ is $(\theta,b)$-derived from the group $(G,\cdot)$. During this paper, we will assume that $(G,f)=\mathrm{der}_{\theta,b}(G,\cdot)$.

Varieties of polyadic groups and the structure of congruences on polyadic groups are studied in [2] and [4]. It is proved that all congruences on polyadic groups are commute and so the lattice of congruences is modular.

There is one more important object associated to polyadic groups. Let $(G,f)$ be a polyadic group. Then, as Post proved, there exists a unique group $(G^{\ast},\circ)$ (which we call now the Post’s cover of $(G,f)$) such that

*1- $G$ is contained in $G^{\ast}$ as a coset of a normal subgroup $R$.

*2- $R$ is isomorphic to a retract of $(G,f)$.

*3- We have $G^{\ast}/R\cong\mathbb{Z}_{n-1}$.

*4- Inside $G^{\ast}$, for all $x_{1},\ldots,x_{n}\in G$, we have $f(x_{1}^{n})=x_{1}\circ x_{2}\circ\cdots\circ x_{n}$.

*5- $G^{\ast}$ is generated by $G$.

The group $G^{\ast}$ is also universal in the class of all groups having properties 1, 4. More precisely, if $\beta:(G,f)\to\mathrm{der}^{n}(H,\ast)$ is a polyadic homomorphism, then there exists a unique ordinary homomorphism $h:G^{\ast}\to H$ such that $h_{|_{G}}=\beta$. This universal property characterizes $G^{\ast}$ uniquely. The explicit construction of the Post’s cover can be find in [19].

3. Free polyadic groups

The structure of free polyadic groups, their construction, and their Post’s cover are studied in [17]. Let $(G,f)=\mathrm{der}_{\theta,b}(G,\cdot)$ be a polyadic group. It is natural to ask when this polyadic group is free. Our main theorem gives a complete solution for this problem. We will need the following well-known theorem of Nielsen-Schrier on the bases of free groups. Note that if $F(X)$ is the free group on a set $X$, $H$ is any subgroup, and $\mathcal{U}$ is a right transversal for $H$, then by the notation $\hat{u}$ we denote the unique element of $\mathcal{U}$ satisfying $Hu=H\hat{u}$. The proof of the following theorem can be find in any standard book of presentation theory of groups.

Theorem 3.1**.**

Let $F(X)$ be the free group of rank $r$, $H$ be a subgroup of index $m$ and $\mathcal{U}$ be a right transversal of $H$. Then the rank of $H$ is equal to $m(r-1)+1$ and the set

[TABLE]

is a basis of $H$.

We can now prove our main theorem.

Theorem 3.2**.**

*Let $(G,f)=\mathrm{der}_{\theta,b}(G,\cdot)$ be a polyadic group. The necessary and sufficient condition for $(G,f)$ to be free of rank $s>1$ is that

1- the group $(G,\cdot)$ is a free of rank $k$ and

[TABLE]

2- there exists a subset $\{v_{1},\ldots,v_{s-1}\}\subseteq G$, such that the set

[TABLE]

is a basis of $(G,\cdot)$.

Proof.

First, assume that $(G,f)$ is free of rank $s>1$ and let

[TABLE]

be a basis. By [17], we have

[TABLE]

where $\mathrm{ht}:F(X)\to\mathbb{Z}$ is the homomorphism $\mathrm{ht}(x_{1}^{\varepsilon_{1}}\ldots x_{m}^{\varepsilon_{m}})=\sum\varepsilon_{i}$.

Suppose that $H=\ker(\mathrm{ht})$. Then we have

[TABLE]

By [1], we know that $F(X)$ is the Post’s cover of $(G,f)$ and since the index of $H$ in $F(X)$ is $n-1$, so $H$ is free of rank $k=(s-1)(n-1)+1$ and by the coset theorem of Post we have $(G,\cdot)\cong H$. So, $(G,f)$ is also a free group of rank $k=(s-1)(n-1)+1$ and this proves 1. Now, consider the right transversal $\mathcal{U}=\{1,u,u^{2},\ldots,u^{n-2}\}$. By the above theorem, the set

[TABLE]

is a basis of $H$. Note that if $x=u$, then

[TABLE]

and hence for $0\leq i\leq n-3$, we have $u^{i}x\widehat{u^{i}x^{-1}}=1$ and for $i-n-2$, we have $u^{i}x\widehat{u^{i}x^{-1}}=u^{n-1}$. If $x=v_{j}$, then

[TABLE]

and since $\mathrm{ht}(u^{i}v_{j})=j+1$, so we have $u^{i}v_{j}\in Hu^{i+1}$. This means that $\widehat{u^{i}v_{j}}=u^{i+1}$. Consequently

[TABLE]

Now, recall that $G=Hu$. By [16] and its notations, the identity of $(G,\cdot)$is $u$ and we have $\overline{w}=w^{2-1n}$, for all $w\in G$. Also, we have

[TABLE]

So, using the relations of Sokolov, we have

[TABLE]

Again, if we use the notations of [16], we have

[TABLE]

and the map $\eta:HtoG$, defined by $\eta(w)=wu$, is an isomorphism. Hence, the set

[TABLE]

is a basis of $(G,\circ)$. Using Nielsen transformations, the next set is also a basis:

[TABLE]

But, since

[TABLE]

so we have

[TABLE]

This proves the assertion 2.

Now, assume that 1 and 2 are true and we prove that

[TABLE]

is free. By 2, suppose $F=F(X)$ and $X=\{u,v_{1},\ldots,v_{s-1}\}$, where $u$ is the identity of $(G,\circ)$. Consider the group $F_{u}$. Inside this group, we have

[TABLE]

We know that $F\cong F_{u}$. So, $F_{u}$ is a free group. Define a homomorphism $\alpha:(G,\circ)\to(F_{u},\ast)$ by

[TABLE]

Put $v_{ij}=\theta^{i}(v_{j})$. For any $m$, we have

[TABLE]

and hence for any $w\in G$, we have

[TABLE]

This shows that for all $w_{1},\ldots,w_{n}\in G$,

[TABLE]

Note that $\alpha(u)=u$ and $\alpha(v_{i})=v_{i}$, and hence $F(X)=\langle\alpha(G)\rangle$. On the other side, $F(X)$ has a normal subgroup $H$, with $H\cong(G,\circ)$, and $[F;H]=m$ such that $m|n-1$. Since $[F_{u};G]=n-1$, so $m=n-1$ and this shows that $F$ is the Post’s cover of $(G,f)$. Now, since $X\subseteq G\subseteq F$, by [1], $(G,f)$ is free.

∎