Subgroups of finite soluble groups inducing the same permutation character

Gavioli, Norberto

doi:10.1090/S0002-9947-97-01696-6

Subgroups of finite soluble groups inducing the same permutation character
HTML articles powered by AMS MathViewer

by Norberto Gavioli PDF

Trans. Amer. Math. Soc. 349 (1997), 2969-2980 Request permission

Abstract:

In this paper there are found necessary and sufficient conditions that a pair of solvable finite groups, say $G$ and $K$, must satisfy for the existence of a solvable finite group $L$ containing two isomorphic copies of $G$ and $H$ inducing the same permutation character. Also a construction of $L$ is given as an iterated wreath product, with respect to their actions on their natural modules, of finite one-dimensional affine groups.

References

A. Caranti, N. Gavioli, and S. Mattarei, Subgroups of finite $p$-groups inducing the same permutation character, Comm. Algebra 22 (1994), no. 3, 877–895. MR 1261011, DOI 10.1080/00927879408824882
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
Walter Feit, Some consequences of the classification of finite simple groups, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 175–181. MR 604576
Robert M. Guralnick, Subgroups inducing the same permutation representation, J. Algebra 81 (1983), no. 2, 312–319. MR 700287, DOI 10.1016/0021-8693(83)90191-6
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Felix Leinen, Chief series and right regular representations of finite $p$-groups, J. Austral. Math. Soc. Ser. A 44 (1988), no. 2, 225–232. MR 922606, DOI 10.1017/S1446788700029803
Robert Perlis, On the equation $\zeta _{K}(s)=\zeta _{K’}(s)$, J. Number Theory 9 (1977), no. 3, 342–360. MR 447188, DOI 10.1016/0022-314X(77)90070-1
Toshikazu Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), no. 1, 169–186. MR 782558, DOI 10.2307/1971195