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Basic subgroups in modular abelian group algebras

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Abstract

Suppose F is a perfect field of char F = p ≠ 0 and G is an arbitrary abelian multiplicative group with a p-basic subgroup B and p-component G p . Let FG be the group algebra with normed group of all units V(FG) and its Sylow p-subgroup S(FG), and let I p (FG; B) be the nilradical of the relative augmentation ideal I(FG; B) of FG with respect to B.

The main results that motivate this article are that 1 + I p (FG; B) is basic in S(FG), and B(1 + I p (FG; B)) is p-basic in V(FG) provided G is p-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston J. Math. when G is p-primary. Thus the problem of obtaining a (p-)basic subgroup in FG is completely resolved provided that the field F is perfect.

Moreover, it is shown that G p (1 + I p (FG; B))/G p is basic in S(FG)/G p , and G(1 + I p (FG; B))/G is basic in V(FG)/G provided G is p-mixed.

As consequences, S(FG) and S(FG)/G p are both starred or divisible groups.

All of the listed assertions enlarge in a new aspect affirmations established by us in Czechoslovak Math. J. (2002), Math. Bohemica (2004) and Math. Slovaca (2005) as well.

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  1. Peter Danchev

    Present address: , 13, General Kutuzov Street, block 7, floor 2, flat 4, 4003, Plovdiv, Bulgaria

Authors and Affiliations

  1. Mathematical Department, Plovdiv State University, 4000, Plovdiv, Bulgaria

    Peter Danchev

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  1. Peter Danchev
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Danchev, P. Basic subgroups in modular abelian group algebras. Czech Math J 57, 173–182 (2007). https://doi.org/10.1007/s10587-007-0053-9

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  • DOI: https://doi.org/10.1007/s10587-007-0053-9

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