To the Post’s coset theorem

A. M. Gal'mak ; N. A. Shchuchkin

Čebyševskij sbornik, Tome 15 (2014) no. 2, pp. 6-20.

Résumé

In the theory of polyadic groups plays an important role groups $A^*$ and $A_0$, appearing in Post's Coset Theorem [2], asserts that for every $n$-ary groups $\langle A, [~] \rangle$ exists a group of $A^*$, in which there is normal subgroup $A_0$ such that the factor group $A^* / A_0$ — cyclic group of order $n-1$. Generator $xA_0$ this cyclic group is the $n$-ary group with $n$-ary operation derived from operation in the group $A^*$, wherein $n$-ary groups $\langle A, [~] \rangle$ and $\langle xA_0, [~] \rangle$ isomorphic. Group $A^*$ is called the Post's universal covering group, and the group $A_0$ — appropriate group. The article begins with a generalization of the Post's Coset Theorem: for every $n$-ary groups $\langle A, [~] \rangle$, $n = k(m-1)+1$, the Post's universal covering group $A^*$ has a normal subgroup $^m \!A$ such that the factor group $A^* / ^m \!A$ — cyclic group of order $m-1$. Moreover, $A_0 \subseteq ~^m \!A \subseteq A^*$ and $^m \!A / A_0$ - cyclic group of order $k$. In this paper we study the permutability of elements in $n$-ary group. In particular, we study the $m$-semi-commutativity in $n$-ary groups, which is a generalization of of the well-known concepts of commutativity and semi-commutativity. Recall that the $n$-ary group $\langle A, [~] \rangle$ is called abelian if it contains any substitution $\sigma$ of the set $\{1,2, \ldots, n \}$ true identity $$ [a_1a_2 \ldots a_n] = [a_{\sigma (1)} a_{\sigma (2)} \ldots a_{\sigma (n)}], $$ and $n$-ary group $\langle A, [~] \rangle$ is called a semi-abelian if it true identity $$ [aa_1 \ldots a_{n-2} b] = [ba_1 \ldots a_{n-2} a]. $$ Summarizing these two definitions, E. Post called $n$-ary group $\langle A, [~] \rangle$ $m$-semi-abelian if $m-1$ divides $n-1$ and $$ (aa_1 \ldots a_{m-2} b, ba_1 \ldots a_{m-2} a) \in \theta_A $$ for any $a, a_1, \ldots, a_{m-2}, b \in A$. We have established a new criterion of $m$-semi-commutativity of $n$-ary group, formulated by a subgroup $^m \!A$ of the Post's universal covering group: $n$-ary group $\langle A, [~] \rangle$ is $m$-semi-abelian if and only if the group $^m \! A$ is abelian. For $n = k(m-1)+1$ by fixed elements $c_1, \ldots, c_{m-2} \in A$ on $n$-ary group of $\langle A, [~] \rangle$ construct $(k+1)$-ary group $\langle A, [~]_{k+1, c_1 \ldots c_{m- 2}} \rangle$. On the coset $A^{(m-1)}$ in generalized Post's Coset Theorem construct $(k+1)$-ary group $\langle A^{(m-1)}, [~]_{k+1} \rangle$. Proved isomorphism of constructed $(k+1)$-ary groups. This isomorphism allows us to prove another criterion $m$-semi-commutativity $n$-ary group: $n$-ary group $\langle A, [~] \rangle$ is $m$-semi-abelian if and only if for some $c_1, \ldots, c_{m-2} \in A$ $(k+1)$-ary group $\langle A, [~]_{k+1, c_1 \ldots c_{m-2}} \rangle$ is abelian. Bibliography: 16 titles.

Détail

Keywords: $n$-ary group, semi-commutativity, coset.

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