Subgroups inducing the same permutation representation. II
In part I [J. Algebra 81, 312-319 (1983; Zbl 0527.20005)] the first author has studied finite groups G containing non-conjugate subgroups H and K with \(ind^ G_ H(I_ H)\cong ind^ G_ K(I_ K)\), the induced representations here being considered as defined over a field of characteristic zero. In this paper the authors first work over the ring \({\mathbb{Z}}_ p\) of p-adic integers. Theorem A: If \([G:H]=[G:K]=p^ a\), then \(ind^ G_ H(I_ H)\cong ind^ G_ K(I_ K)\) as \({\mathbb{Z}}_ pG\)-modules, iff \(P\cap H\) and \(P\cap K\) are conjugate in G, where P is a Sylow p-subgroup of G. The theorem is used to strengthen a result of \textit{R. Perlis}' [J. Number Theory 10, 489-509 (1978; Zbl 0393.12009)], namely that number fields E and F of degree \(p^ a\) over \({\mathbb{Q}}\) and with the same normal hull H have the same zeta function as well as the same p-part of their class numbers provided that \(p^{a+1}\nmid [EF^{\sigma}:{\mathbb{Q}}]\) for all \(\sigma\in Gal(H/{\mathbb{Q}})\). Theorem B: Assume \([G:H]=pq\), a product of two distinct primes. Then, if \(ind^ G_ H(I_ H)\cong ind^ G_ K(I_ K)\) as \({\mathbb{Q}}G\)-modules, one of the following holds: a) H and K are conjugate (and this is necessarily the case when G is solvable), b) p,q or \(pq\in \{11\}\cup (r^ n-1)/(r- 1)\), \(n\geq 3\), r a prime power\(\}\), c) q (say) is a Fermat prime and \(q\equiv 2 mod p\), d) \(pq=29\cdot 59\). The proof is based on the classification of simple groups. Many examples are worked out.
scientific article; zbMATH DE number 3922935
scientific article