Presentations of metacyclic \(p\)-groups with applications to \(K\)- admissibility questions
Let \(p\) be a prime, \(G\) a finite metacyclic \(p\)-group. Then \(G=\langle x, y\mid x^{p^ n}=e\), \(y^{p^ m}=x^{p^ s}\), \(yxy^{-1}=x^ q\rangle\) where \(q=(p+1)^{p^ t}\) if \(p\) is odd, \(q=s^{2^ t}\) if \(p =2\) and \(q \equiv 1\pmod 4\), and \(q =-5^{2^ t}\) if \(p=2\) and \(q \equiv 3 \pmod 4\). For fixed \(p\), the parameters of \(G\) are \((n, m, t, s)\). Let \(Z(G)\) be the center of \(G\) and \(G^{ab}\) its commutator quotient. Let \(H\) be another metacyclic \(p\)-group with parameters \((n', m', t', s')\). Assume \(G\) and \(H\) have the same order. The author proves: (1) If \(p\) is odd or \(p=2\) and \(q \equiv 1 \pmod 4\), then \(G \cong H\) iff \(Z(G) \cong Z(H)\) and \(G^{ab} \cong H^{ab}\). (2) If \(p\) is odd, then \(G \cong H\) iff \(m+t=m'+t'\), \(s=s'\), and \(\min(s,t+1)=\min(s',t'+1)\). (3) If \(p=2\) and \(q \equiv 1 \pmod 4\), then \(G \cong H\) iff \(m+t=m'+t'\), \(s=s'\), and \(\min(s,t+2)=\min(s', t'+2)\). (3) If \(p=2\) and \(q \equiv 3 \pmod 4\), then \(G \cong H\) iff either (i) \(G\) and \(H\) have the same parameters, or (ii) \(n=n'\), \(m=m'\), \(n \geq 3\), \(m \geq 2\), \(t=n - 3\), \(t'=n - 2\), and \(s=s'=n - 1\).
scientific article; zbMATH DE number 701289