We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented $$\begin{aligned} G(m,n,k) = \left\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad (m,n,k\in {\mathbb {Z}}^+) \end{aligned}$$ G ( m , n , k ) = a , b ; a m = 1 , b n = 1 , a b = a k ( m , n , k ∈ Z + ) with $$(m,k - 1) = 1$$ ( m , k - 1 ) = 1 and $$n = \mathrm {ind}_m(k),$$ n = ind m ( k ) , the smallest positive integer for which $${k^n} = 1\,\pmod m,$$ k n = 1 ( mod m ) , with the conjugate of a by b written $${a^b} = {b^{ - 1}}ab.$$ a b = b - 1 a b . The right and left commutation semigroups of G , denoted $$\mathrm{P}(G)$$ P ( G ) and $$\Lambda (G),$$ Λ ( G ) , are the semigroups of mappings generated by $$\rho (g):G \rightarrow G$$ ρ ( g ) : G → G and $$\lambda (g):G \rightarrow G$$ λ ( g ) : G → G defined by $$(x)\rho (g) = [x,g]$$ ( x ) ρ ( g ) = [ x , g ] and $$(x)\lambda (g) = [g,x],$$ ( x ) λ ( g ) = [ g , x ] , where the commutator of g and h is defined as $$[g,h] = {g^{ - 1}}{h^{ - 1}}gh.$$ [ g , h ] = g - 1 h - 1 g h . This paper builds on a previous study of commutation semigroups of dihedral groups conducted by the authors with C. Levy. Here we show that a similar approach can be applied to G , a metacyclic group with trivial centre. We give a construction of $$\mathrm{P}(G)$$ P ( G ) and $$\Lambda (G)$$ Λ ( G ) as unions of containers , an idea presented in the previous paper on dihedral groups. In the case that $$\left\langle a \right\rangle$$ a is cyclic of order p or $${p^2}$$ p 2 or its index is prime, we show that both $$\mathrm{P}(G)$$ P ( G ) and $$\Lambda (G)$$ Λ ( G ) are disjoint unions of maximal containers. In these cases, we give an explicit representation of the elements of each commutation semigroup as well as formulas for their exact orders. Finally, we extend a result of J. Countryman to show that, for G ( m , n , k ) with m prime, the condition $$\left| {\mathrm{P}(G)} \right| = \left| {\Lambda (G)} \right|$$ P ( G ) = Λ ( G ) is equivalent to $$\mathrm{P}(G) = \Lambda (G).$$ P ( G ) = Λ ( G ) .

中文翻译：

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具有平凡中心的有限元循环群的对易半群

我们研究了具有平凡中心的有限元循环群的左右对易半群。这些被呈现 $$\begin{aligned} G(m,n,k) = \left\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad (m,n,k\in {\mathbb {Z}}^+) \end{aligned}$$ G ( m , n , k ) = a , b ; am = 1 , bn = 1 , ab = ak ( m , n , k ∈ Z + ) 其中 $$(m,k - 1) = 1$$ ( m , k - 1 ) = 1 且 $$n = \ mathrm {ind}_m(k),$$ n = ind m ( k ) ，$${k^n} = 1\,\pmod m,$$ kn = 1 ( mod m ) 的最小正整数， a 与 b 的共轭写为 $${a^b} = {b^{ - 1}}ab.$$ ab = b - 1 ab 。G 的左右对易半群，记为 $$\mathrm{P}(G)$$ P ( G ) 和 $$\Lambda (G),$$ Λ ( G ) ，是由 $ 生成的映射的半群$\rho (g):G \rightarrow G$$ ρ ( g ) : G → G 和 $$\lambda (g):G \rightarrow G$$ λ ( g ) ：G → G 由 $$(x)\rho (g) = [x,g]$$ ( x ) ρ ( g ) = [ x , g ] 和 $$(x)\lambda (g) = [g ,x],$$ ( x ) λ ( g ) = [ g , x ] ，其中 g 和 h 的交换子定义为 $$[g,h] = {g^{ - 1}}{h^{ - 1}}gh.$$ [ g , h ] = g - 1 h - 1 gh 。本文建立在作者与 C. Levy 先前对二面体群的对易半群的研究之上。在这里，我们展示了类似的方法可以应用于 G ，一个具有平凡中心的元循环群。我们给出了 $$\mathrm{P}(G)$$ P ( G ) 和 $$\Lambda (G)$$ Λ ( G ) 的构造作为容器的并集，这是之前关于二面体群的论文中提出的想法. 在 $$\left\langle a \right\rangle$$ a 是 p 阶循环或 $${p^2}$$ p 2 或其索引为素数的情况下，我们证明 $$\mathrm{ P}(G)$$ P ( G ) 和 $$\Lambda (G)$$ Λ ( G ) 是最大容器的不相交联合。在这些情况下，我们给出了每个对易半群的元素的明确表示以及它们的确切阶数的公式。最后，我们扩展 J. Countryman 的结果来证明，对于 G ( m , n , k ) 与 m 个素数，条件 $$\left| {\mathrm{P}(G)} \right| = \左| {\Lambda (G)} \right|$$ P ( G ) = Λ ( G ) 等价于 $$\mathrm{P}(G) = \Lambda (G)。$$ P ( G ) = Λ ( G ） 。