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Brief communications
Commutativity of involutive twovalued groups
A. A. Gaifullin^{abcd}^{document.write(decode_email('DMEBCAHEGJHEGMGFDNCHEFCNGNGBGJGMDKCAGBGHGBGJGGEAGNGJCNHCGBHDCOHCHFCHCAGDGMGBHDHDDNFDEMGJGOGLCAEIFCEFEGDNCCGNGBGJGMHEGPDKGBGHGBGJGGEAGNGJCNHCGBHDCOHCHFCCDODMGJGNGHCAHDHEHJGMGFDNCCGNGBHCGHGJGOCNGMGFGGHEDKDDHAHICCCAGBGMGJGHGODNCCGBGCHDGNGJGEGEGMGFCCCAHDHCGDDNCCCPGGHEGJGDGPGOHDCPGFGNGBGJGMGJGDGPDBCOGKHAGHCCCAHHGJGEHEGIDNCCDBDIHAHICCCAGCGPHCGEGFHCDNCCDACCCPDODMCPEBDO'));email} ^{a} Steklov Mathematical Institute of Russian Academy of Sciences
^{b} Skolkovo Institute of Science and Technology
^{c} Lomonosov Moscow State University
^{d} Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
Received: 25.01.2024
Over the past decades, the theory of $n$valued groups has been enriched by numerous new results and applications; see [1] and [2]. One of its features is the complexity of classification problems even in the commutative case. In [2] Buchstaber and Veselov, in connection with the Conway topograph, introduced the class of involutive twovalued groups, for which the classification problems seem amenable: see [3]. Following [2] and [3], we give the following definition.
Definition. An involutive twovalued group is a set $X$ endowed with a multiplication $*\colon X\times X\to\operatorname{Sym}^2(X)$ (where $\operatorname{Sym}^2(X)$ is the symmetric square of $X$) and an identity element $e\in X$ that have the following properties: (1) (associativity) for any $x,y,z\in X$ there is an equality of $4$element multisets $(x*y)*z=x*(y*z)$, (2) (strong identity) $x*e=e*x=[x,x]$ for all $x\in X$, (3) (involutivity) the multiset $x*y$ contains the identity $e$ if and only if $x=y$. A twovalued group is said to be commutative if it, in addition, has the following property: (4) (commutativity) $x*y=y*x$ for all $x,y\in X$.
Note that in [4] the concept of an involutive $n$valued group was used in another, not equivalent meaning: see Remark 1.5 in [3] for details.
In [3] a complete classification of finitely generated commutative involutive two valued groups and partial classification results in the nonfinitely generated and topological cases were obtained. The main result of the present note is the following theorem, which implies that all classification results in [3] are also valid without the assumption of commutativity.
Theorem. Any involutive twovalued group is commutative.
For singlegenerated twovalued groups this theorem was proved in [3]. The following lemma is Lemma 2.2 in [3].
Lemma 1. Suppose that $x$, $y$, and $z$ are elements of an involutive twovalued group $X$. Then $z\in x*y$ if and only if $y\in z*x$.
Of key importance for us will be the concepts of powers and order of an element of an involutive twovalued group, which were introduced in [3]. Namely, for each element $x\in X$ there is a unique sequence $x^0=e$, $x^1=x$, $x^2,x^3,\ldots$ of elements of $X$ such that $x^k*x^m=[x^{km},x^{k+m}]$ for all $k$ and $m$. The order of $x$ (denoted by $\operatorname{ord} x$) is the smallest positive integer $k$ such that $x^k=e$. In particular, the order of a nonidentity element $x$ is equal to $2$ if and only if $x*x=[e,e]$.
Lemma 2. Suppose that $x$ and $y$ are elements of an involutive twovalued group $X$ such that $\operatorname{ord} x =2$. Then $x*y=y*x=[z,z]$ for some element $z\in X$.
Proof. Let $z$ be an element of $x*y$. By Lemma 1 the multiset $z*x$ contains $y$, that is, $z*x=[y,y']$ for some $y'$. Then $[y*x,y'*x]=(z*x)*x=z*(x*x)=z*[e,e]=[z,z,z,z]$. Hence $y*x=[z,z]$. Applying Lemma 1 again, we obtain $x*z=[y,y'']$ for some $y''$. Then $[x*y,x*y'']=x*(x*z)=(x*x)*z=[e,e]*z=[z,z,z,z]$. Therefore, $x*y=[z,z]$.
Proof of the theorem. We need to prove that $x*y=y*x$ for all $x$ and $y$. If one of the two elements $x$ and $y$ has order $2$, then the required equality follows from Lemma 2. So we may assume that $\operatorname{ord} x>2$ and $\operatorname{ord} y>2$, that is, $x^2\ne e$ and $y^2\ne e$. Put $x*y=[z_1,z_2]$ and $y*x=[w_1,w_2]$. Then we have
$$
\begin{equation}
\begin{aligned} \, \nonumber [z_1*w_1,z_1*w_2,z_2*w_1,z_2*w_2]&=(x*y)*(y*x)=x*(y*y)*x \\ &=x*[e,y^2]*x=[e,e,x^2,x^2,x*y^2*x]. \end{aligned}
\end{equation}
\tag{1}
$$
Hence either at least two of the four multisets $z_i*w_j$ contain $e$ or one of these four multisets is equal to $[e,e]$. From involutivity it follows that $e\in z_i*w_j$ if and only if $z_i=w_j$. Up to the permutations $z_1\leftrightarrow z_2$ and $w_1\leftrightarrow w_2$ and the reversal of the roles of the elements $x$ and $y$, there are three substantially different cases.
Case 1: $z_1=w_1$ and $z_2=w_2$. Then $x*y=y*x$, as required.
Case 2: $z_1=z_2=w_1\ne w_2$. We have
$$
\begin{equation}
x*(y*z_1)=(x*y)*z_1=[z_1,z_1]*z_1=[e,e,z_1^2,z_1^2].
\end{equation}
\tag{2}
$$
Since $\operatorname{ord} x>2$, we have $x*x=[e,x^2]$, where $x^2\ne e$. Moreover, by involutivity $e\notin x*x'$ unless $x'=x$. Hence it follows from (2) that $y*z_1=[x,x]$ and $x^2=z_1^2$. It is proved similarly that $z_1*x=[y,y]$ and $y^2=z_1^2$. Therefore, $x^2=y^2$. Consequently, $z_1*w_1=z_2*w_1=z_1*z_1=[e,z_1^2]=[e,x^2]$ and $x*y^2*x=x*x^2*x=[x,x^3]*x=[e,x^2,x^2,x^4]$. Thus, (1) reads $[e,e,x^2,x^2,z_1*w_2,z_1*w_2]=[e,e,e,x^2,x^2,x^2,x^2,x^4]$. Since $x^2\ne e$, we obtain $e\in z_1*w_2$, which leads to a contradiction since $z_1\ne w_2$. Therefore, the case under consideration is impossible
Case 3: $z_1*w_1=[e,e]$, that is, $z_1=w_1$ and $\operatorname{ord} z_1=2$. We may assume that the elements $z_1$, $z_2$, and $w_2$ are pairwise different, since otherwise we arrive at one of the two cases already considered (Case 1 or 2). By Lemma 1 we obtain $y\in z_1*x$. Since $\operatorname{ord} z_1=2$, it follows from Lemma 2 that $x*z_1=z_1*x=[y,y]$. We have
$$
\begin{equation*}
(x*z_1)*(z_1*x)=[y,y]*[y,y]=[e,e,e,e,y^2,y^2,y^2,y^2]
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
x*(z_1*z_1)*x=x*[e,e]*x=[e,e,e,e,x^2,x^2,x^2,x^2].
\end{equation*}
\notag
$$
Hence $x^2=y^2$. As in Case 2, it follows that $e\in x*y^2*x$, and therefore $e$ occurs in the multiset (1) with multiplicity at least three. Hence $e$ is contained in at least one of the three multisets $z_1*w_2$, $z_2*w_1=z_2*z_1$, and $z_2*w_2$, which is impossible, since the elements $z_1$, $z_2$, and $w_2$ are pairwise different. The contradiction obtained completes the proof of the theorem.
The author is grateful to V. M. Buchstaber and A. P. Veselov for multiple fruitful discussions.



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Citation:
A. A. Gaifullin, “Commutativity of involutive twovalued groups”, Uspekhi Mat. Nauk, 79:2(476) (2024), 185–186; Russian Math. Surveys, 79:2 (2024), 363–365
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https://www.mathnet.ru/eng/rm10170https://doi.org/10.4213/rm10170e https://www.mathnet.ru/eng/rm/v79/i2/p185

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