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Basic subgroups in modular abelian group algebras
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 173-182 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Suppose ${F}$ is a perfect field of ${\mathop {\mathrm char}F=p\ne 0}$ 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.
Suppose ${F}$ is a perfect field of ${\mathop {\mathrm char}F=p\ne 0}$ 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.
Classification : 16S34, 16U60, 20K10, 20K20, 20K21
Keywords: $p$-basic subgroups; normalized units; group algebras; starred groups 
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     title = {Basic subgroups in modular abelian group algebras},
     journal = {Czechoslovak Mathematical Journal},
     pages = {173--182},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a14/}
}
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Danchev, Peter. Basic subgroups in modular abelian group algebras. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 173-182. http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a14/

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