Yilun Shang: Groups with $(m_1,m_2,n)$-permutational property, p.319-325

Abstract:

Let $G$ be a group, $\mathcal{A}$ and $\mathcal{B}$ be two sets of $n$-tuples of elements of $G$ with $\vert\mathcal{A}\vert=m_1$m₁ and $\vert\mathcal{B}\vert=m_2$m₂, respectively. $G$ is said to have the $(m_1,m_2,n)$-permutational property with respect to $\mathcal{A}$ and $\mathcal{B}$ if for all elements $g_1,g_2,\cdots,g_n\in G$, there exist $a_1,a_2,\cdots,a_n\in \mathcal{A}$, $b_1,b_2,\cdots,b_n\in \mathcal{B}$ and a nonidentity permutation $\sigma\in Sym_n$ such that

$$a_1g_1b_1a_2g_2b_2\cdots a_ng_nb_n=a_{\sigma(1)}g_{\sigma(1)}... ...2)}b_{\sigma(2)}\cdots a_{\sigma(n)}g_{\sigma(n)}b_{\sigma(n)}.$$

We show that if $G$ is $(m_1,m_2,n)$-permutational, then $G$ has a characteristic subgroup $N$ such that $\vert G:N\vert$ and $\vert N'\vert$ are both finite and have sizes bounded by functions of $m_1,m_2$ and $n$. As a consequence, if $\Delta$ is the finite conjugate center of the group, then $\vert G:\Delta\vert$ and $\Delta'$ are both finite with $\vert G:\Delta\vert$ bounded by a function of $m_1,m_2$ and $n$.

Key Words: Group, permutational property, finite conjugate center.

2000 Mathematics Subject Classification: Primary: 20E25;
Secondary: 20B30.

Download the paper in pdf format here.