Let \(p\) be a prime, \(e\), \(m\) positive integers, \(P=C\wr E\) the wreath product of the cyclic group \(C\) of order \(p^e\), \(E\) elementary abelian of order \(p^m\). In his classical article H. Liebeck [Proc. Camb. Philos. Soc. 58, 443-451 (1962; Zbl 0106.24902)] determined the nilpotency class of \(P\) as \((e+m-1)(p-1)+1\). In the present paper the author determines the upper central series, this way describing an important aspect of the structure of the wreath product, continuing his research initiated in [J. M. Riedl, Commun. Algebra 43, No. 5, 2152-2173 (2015; Zbl 1328.20051)].
In order to state the main result, some technicalities are due. Let \(B\) be the product of \(p^m\) copies of \(C\) with generators \(x_u\) with \(u\) in \(\mathcal U=\{0,1,\ldots,p-1\}^m\). For \(1\leq j\leq m\) by letting \(y_j\colon x_u\mapsto x_{u'}\), \(u'\) is obtained from \(u\) by decreasing the \(j\)-th coordinate by 1, we get an automorphism of \(B\), let \(Y\) be the group generated by the automorphisms \(y_j\), then \(P\) is isomorphic to the semidirect product of \(B\) by \(Y\). Let \(\mathcal F\) be the set of all functions \(\mathcal U\to\mathbb Z/p^e\mathbb Z\), it is an additive group, denote by \(x(f)\) the product of all \(x_u^{f(u)}\). The mapping \(f\mapsto x(f)\) is an isomorphism of \(\mathcal F\) and \(B\). Denote by \({u\brack v}\) the product all \(u_i\choose v_i\) modulo \(p^e\) and let \(e_u\in\mathcal F\) be defined by \(e(u)={u\brack v}\). Then any \(f\) can be written as \(\sum_u\widehat f(u)e_u\), and the map defined by \(f\mapsto\widehat f\) is an automorphism of \(\mathcal F\). Let \(\mathcal P\) denote the set of all functions \(\mathcal U\to\{0,1,\ldots,e\}\), called patterns. To every pattern \(\alpha\) belongs a subgroup \(S(\alpha)\), called a pattern subgroup, of \(B\) generated by all \(x(f)\) with \(\widehat f(u)\) contained in the unique subgroup of \(\mathbb Z/p^e\mathcal Z\) of order \(\alpha(u)\) for any \(u\). Let \(\omega_0\colon\mathbb Z\to\{0,e\}\) be defined by \(\omega_0(k)=0\) if \(k\leq 0\), \(e\) otherwise; \(\theta(k)\) by \(\theta(k)=0\) for \(k\leq 0\), \(\theta(k)=k\) for \(0<k<e\), \(e\) otherwise; \(\omega_1(k)=\theta(\lceil k/(p-1)\rceil)\); for \(u\in\mathcal U\) let \(\sigma(u)\) be the sum of the coordinates; let \(\varepsilon\in\mathcal U\) denote the tuple with all coordinates zero; finally, let the pattern \(\eta_i\) be defined by \(\eta_i(u)=\omega_0(i)\) if \(u=\varepsilon\) and \(\eta_i(u)=\omega_1(i-\sigma(u))\).
The main result now can be stated briefly, the proper term \(\zeta_i\) of the upper central series of \(P\) is just \(S(\eta_i)\).