Rewritable groups. (Q408512)

A group \(G\) is said to have the \(n\)-rewritable property \(Q_n\) if for all elements \(g_1,g_2,\dots,g_n\in G\), there exist two distinct permutations \(\sigma\) and \(\tau\in\text{Sym}_n\) such that NEWLINE\[NEWLINEg_{\sigma(1)}g_{\sigma(2)}\cdots g_{\sigma(n)}=g_{\tau(1)}g_{\tau(2)}\cdots g_{\tau(n)}.NEWLINE\]NEWLINE For a group \(G\), \(\Delta(G)\) is the set of elements \(g\in G\) whose centralizers in \(G\) have finite index. \(\Delta(G)\) is a characteristic subgroup of \(G\).NEWLINENEWLINE The main results of the paper under review are the following:NEWLINENEWLINE Theorem 1.1. Let \(G\) be a group satisfying the rewritable property \(Q_n\). Then \(G\) has a characteristic subgroup \(N\), contained in \(\Delta(G)\), such that \(|G:N|\) and \(|N'|\) are finite and have sizes bounded by functions of \(n\).NEWLINENEWLINE A group \(G\) is said to have the \(n\)-permutational property \(P_n\) if for all elements \(g_1,g_2,\dots,g_n\in G\), there exists a nonidentity permutation \(\sigma\in\text{Sym}_n\) such that NEWLINE\[NEWLINEg_1g_2\cdots g_n=g_{\sigma(1)}g_{\sigma(2)}\cdots g_{\sigma(n)}.NEWLINE\]NEWLINE It is clear that \(Q_n\subseteq P_n\) for all \(n>1\). The following result shows that there exists a function \(f\colon\mathbb N\to\mathbb N\) such that \(P_n\subset Q_{f(n)}\).NEWLINENEWLINE Corollary 1.3. If \(G\) satisfies the rewritable property \(Q_n\), then \(G\) satisfies the permutational property \(P_m\) with \(m\) bounded by a function of \(n\).NEWLINENEWLINE The authors of the paper under review introduce a generalization of \(n\)-permutational groups as follows. Let \(m\) and \(n\) be positive integers and suppose that \(\mathcal A\) is a set of \(n\)-tuples \((a_1,a_2,\dots,a_n)\) of elements of \(G\) with \(|\mathcal A|=m\). A group \(G\) is said to be \((m,n)\)-permutational with respect to \(\mathcal A\) if for every \(n\)-tuple \((x_1,x_2,\dots,x_n)\) of elements of \(G\) there exist an \(n\)-tuple \((a_1,a_2,\dots,a_n)\in\mathcal A\) and a nonidentity permutation \(\sigma\in\text{Sym}_n\) such that NEWLINE\[NEWLINEx_1a_1x_2a_2\cdots x_na_n=x_{\sigma(1)}a_{\sigma(1)}x_{\sigma(2)}a_{\sigma(2)}\cdots x_{\sigma(n)}a_{\sigma(n)}.NEWLINE\]NEWLINE For a group \(G\) and each integer \(k\), \(\Delta_k(G)\) denotes the set \(\{x\in G\mid |G:C_G(x)|\leq k\}\).NEWLINENEWLINE The proof of the following result uses arguments as that of Theorem 2.6 of the paper under review.NEWLINENEWLINE Proposition 4.1. If \(G\) is an \((m,n)\)-permutational group with respect to \(\mathcal A\), then setting \(k=m\cdot n!\), we haveNEWLINENEWLINE (i) \(|G:\Delta_k(G)|\leq k\cdot(k+1)!\), andNEWLINENEWLINE (ii) \(G\) has a characteristic subgroup \(N\) with \(|G:N|\leq k\cdot (k+1)!\), and with \(|N'|\) finite and bounded by a function of \(n\).NEWLINENEWLINE In the last section of the paper under review the authors discuss known results on group algebras satisfying a polynomial identity. Let \(K\) be a field and let \(\mathcal F=K\langle\zeta_1,\zeta_2,\zeta_3,\dots\rangle\) be the free \(K\)-algebra in the noncommuting variables \(\zeta_1,\zeta_2,\zeta_3,\dots\). A \(K\)-algebra \(R\) is said to satisfy the polynomial identity \(f(\zeta_1,\zeta_2,\dots,\zeta_k)\in\mathcal F\) if \(f(r_1,r_2,\dots,r_k)=0\) for all \(r_1,r_2,\dots,r_k\in R\).NEWLINENEWLINE The main similarity between algebras satisfying a polynomial identity and permutational groups is perhaps Lemma 5.1 of the paper saying that every \(K\)-algebra \(R\) satisfying a polynomial identity of degree \(n\), satisfies a multilinear identity of the form NEWLINE\[NEWLINEf(\zeta_1,\zeta_2,\dots,\zeta_n)=\sum_{\sigma\in\text{Sym}_n} k_\sigma\zeta_{\sigma(1)}\zeta_{\sigma(2)}\cdots\zeta_{\sigma(n)},NEWLINE\]NEWLINE with \(k_\sigma\in K\) and \(k_1=1\).