The Normalizer Property for Integral Group Rings of Complete Monomial Groups

Abstract

Let G be a complete monomial group with nilpotent base, namely, G = NwrSymm, the wreath product of a finite nilpotent group N with the symmetric group on m letters. Then G satisfies the normalizer conjecture for ZG.

Coleman Automorphisms of Permutational Wreath Products

“…Petit Lobão and Sehgal (see [22,Theorem]) established that the normalizer property holds for the natural wreath product G = N S n , where N is a nilpotent group and S n is the symmetric group of degree n. Observing that natural wreath products are special cases of permutational wreath products and S n is centerless whenever n = 2, we shall extend their result to a more general setting. Our main result is as follows.…”

“…Hertweck [2] constructed a group G of order 2 25 · 97 2 for which the normalizer property fails. For positive results on the normalizer problem, the reader may refer to [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. According to the Coleman's result ([7, Coleman Lemma]), the normalizer property holds for G provided that Out Col G = 1.…”

The Isomorphism Problem for Integral Group Rings of a Complete Monomial Group

“…Similar groups have been analyzed in the literature of group rings: the (Aut) conjecture of Zassenhaus was established for wreath products AwrSym m , PwrSym m , and Sym n wrSym m in Giambruno et al (1990), Giambruno and Sehgal (1992), Parmenter andValenti (1992). Petit Lobão and Sehgal (2003) also proved the normalizer conjecture (Nor)-which has an interesting 4408 PETIT LOBÃO relation with (Iso), as observed in Mazur (1995)-to a complete monomial group with a nilpotent group on the bottom. In our main result, we shall prove that groups which are extensions of an abelian group by a symmetric group on m letters through a wreath action are determined by their integral group rings.…”

Coleman automorphisms of generalized dihedral groups

“…Thus it is no surprise that Coleman automorphisms of G occur naturally in the study of the normalizer problem. Related work in this direction can be found in [3][4][5][6][7][8][9][10][11][12]. However, we do not intend to go into details for this.…”