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Chebyshevskii Sbornik, 2014, Volume 15, Issue 2, Pages 6–20 (Mi cheb337)  

To the Post’s coset theorem

A. M. Gal'maka, N. A. Shchuchkinb

a Mogilev State Foodstaffs University
b Volgograd State Socio-Pedagogical University
References:
Abstract: 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.
Keywords: $n$-ary group, semi-commutativity, coset.
Received: 19.05.2014
Document Type: Article
UDC: 512.548
Language: Russian
Citation: A. M. Gal'mak, N. A. Shchuchkin, “To the Post’s coset theorem”, Chebyshevskii Sb., 15:2 (2014), 6–20
Citation in format AMSBIB
\Bibitem{GalShc14}
\by A.~M.~Gal'mak, N.~A.~Shchuchkin
\paper To the Post’s coset theorem
\jour Chebyshevskii Sb.
\yr 2014
\vol 15
\issue 2
\pages 6--20
\mathnet{http://mi.mathnet.ru/cheb337}
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