Basic subgroups in modular abelian group algebras

Danchev, Peter Vassilev. "Basic subgroups in modular abelian group algebras." Czechoslovak Mathematical Journal 57.1 (2007): 173-182. <http://eudml.org/doc/31122>.

@article{Danchev2007,
abstract = {Suppose $\{F\}$ is a perfect field of $\{\mathop \{\mathrm \{c\}har\}F=p\ne 0\}$ 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.},
author = {Danchev, Peter Vassilev},
journal = {Czechoslovak Mathematical Journal},
keywords = {$p$-basic subgroups; normalized units; group algebras; starred groups; -basic subgroups; normalized units; group algebras; starred groups; groups of units; augmentation ideals},
language = {eng},
number = {1},
pages = {173-182},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Basic subgroups in modular abelian group algebras},
url = {http://eudml.org/doc/31122},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Danchev, Peter Vassilev
TI - Basic subgroups in modular abelian group algebras
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 173
EP - 182
AB - Suppose ${F}$ is a perfect field of ${\mathop {\mathrm {c}har}F=p\ne 0}$ 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.
LA - eng
KW - $p$-basic subgroups; normalized units; group algebras; starred groups; -basic subgroups; normalized units; group algebras; starred groups; groups of units; augmentation ideals
UR - http://eudml.org/doc/31122
ER -