Computing \(p\)-groups with trivial Schur multiplicator.
The author continues her investigation of \(p\)-groups with trivial Schur multiplicator. She has previously proved [Isr. J. Math. 166, 157-166 (2008; Zbl 1153.20015)] that, for odd primes, there are only finitely many \(p\)-groups with trivial Schur multiplicator of a given coclass. (Recall a \(p\)-group \(G\) of order \(p^n\) has coclass \(r\) if \(r=n-c\) and \(c\) is the nilpotency class of \(G\).) The author introduces an algorithm to compute these finitely many \(p\)-groups for a given prime and coclass. The author classifies all such groups of coclass 1, gives a conjectural description for them for coclass 2 and determines them for coclass 3 and \(p\in\{3,5,7\}\) and coclass 4 and \(p=3\). The algorithm computes Schur towers. A nilpotent group \(G\) is a Schur tower if every lower central series factor \(G/\gamma_{j+1}(G)\) is a Schur cover of the factor \(G/\gamma_j(G)\). Every nilpotent group with trivial Schur multiplicator is a Schur tower and furthermore, for odd \(p\), there are only finitely many Schur towers among the \(p\)-groups of coclass \(r\). A main part of the algorithm is an adaptation of an algorithm of \textit{W. Nickel} [PhD thesis, ANU, Canberra (1993)] which computes the isomorphism types of Schur covers of finite \(p\)-groups. The author is led to conjecture that, for \(p\) odd, the order of a \(p\)-group with trivial Schur multiplicator and coclass \(r\) is bounded by a function \(b(r)\) which is independent of \(p\). \textit{M. F. Newman} has confirmed this for coclass 2 [J. Algebra 322, No. 3, 910-913 (2009; Zbl 1196.20020)].
scientific article; zbMATH DE number 5610502
scientific article