_09_Motose - Math. J. Okayama Univ. 44 (2002), 5156 ON THE...

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Unformatted text preview: Math. J. Okayama Univ. 44 (2002), 5156 ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI Kaoru MOTOSE Let t ( G ) be the nilpotency index of the radical J ( KG ) of a group algebra KG of a finite p-solvable group G over a field K of characteristic p > 0. Then it is well known by D. A. R. Wallace [7] that p e t ( G ) e ( p- 1) + 1 , where p e is the order of a Sylow p-subgroup of G . H. Fukushima [1] characterized a group G of p-length 2 satisfying t ( G ) = e ( p- 1) + 1, see also [4]. Unfortunately, his characterization holds under a condition such that the p-part V = O p ,p ( G ) /O p ( G ) of G is abelian. In this paper, using Dickson near fields, we shall give an explicit example (see Example 1) such that a group G of p-length 2 has the non abelian p- part V and satisfies t ( G ) = e ( p- 1)+1. This example will be new and have a contributions in our research. Example 2 is also very interesting because quite different objects (see [3] and [5]) are unified on the ground of Dickson near fields. Let H be a sharply 2-fold transitive group on = { , 1 ,,,..., } (see [8, p. 22]). Let V = H be a stabilizer of 0, and let U be the set consisting of the identity and fixed point-free permutations in H . Then U is an elementary abelian p-subgroup of H with the order p s (see Lemma 1). Let be a permutation of order p on satisfying conditions H- 1 H, p = 1 , (0) = 0 and (1) = 1 . Then it is easy to see U- 1 U and V - 1 V . We set W = h i and C V ( ) = { v V | v = v } . Assume that there exists a normal subgroup T of WV contained in V such that V is a semi-direct product of T by C V ( ). We set G = h W,T,U i . Now, we present the well known results Lemmas 1 and 2 for completeness of this paper. Lemma 1. U is a normal and elementary abelian p-subgroup of H . Proof. First we shall prove, for k * = \ { } , there exists only one u k U with u k (0) = k , equivalently, the following map from U to is bijective: : u u (0) . This paper was financially supported by the Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (Subject No. 1164003). 51 52 K. MOTOSE For U \{ } , there exists H with ( (0)) = k since (0) 6 = 0 and H is transitive on * . We set u k = - 1 . Then u k U and u k (0) = k . Thus is surjective. It follows from definition of H and U that U = H \ a ( H a \ { } ) , ( H a \ { } ) ( H b \ { } ) = for a 6 = b. Using | H | = | H a || a H | = | H a || | , where a H is an orbit of a , we can see | U | = | | . Hence is injective. Assume has a fixed point for , U . Then we may assume = 0 since H is transitive on and U- 1 = U for H . Thus = - 1 follows from - 1 U , (0) = - 1 (0) and the above observation. This means U . Hence U is a normal subgroup of H because U- 1...
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_09_Motose - Math. J. Okayama Univ. 44 (2002), 5156 ON THE...

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