Timeline for Radical of group algebra

6 events

when toggle format	what		by	license	comment
Jan 1, 2012 at 12:23	comment	added	Salvatore Siciliano		Benjamin: the case in which G has a normal p-Sylow subgroup is known as Wallace Theorem.
Dec 31, 2011 at 23:26	answer	added	Geoff Robinson		timeline score: 2
Sep 27, 2011 at 18:30	comment	added	Benjamin Steinberg		When G has a normal p-Sylow subgroup P, then the group homomorphism $G\rightarrow G/P$ induces the semisimple quotient at the level of algebras (since $F_p[G/P]$ is semisimple by Maschke's theorem). So in this case the radical is generated by all elements of $F_p[G]$ such that the coefficients of the elements of $P$ sum to zero. In general, $G$ has a largest normal $P$-subgroup $N$, but $G/N$ is not semisimple if $N$ is not $p$-Sylow. Thus $F_p[G]\right F_p[G/N]$ is not the semisimple quotient. Clearly though it suffices to compute $rad(F_p[G/N])$ so you are reduced to the case $N$ is trivial
Sep 27, 2011 at 18:18	comment	added	student		Thanks for reference. For example when G have normal p-group P then I know simple answer. In this case as I understand radical is $Ind_P^G\Delta(P)$, where $Delta(P)$ are all elements of $F_p[P]$ with sum zero. Does there exists similar answer when P is not normal subgroup? – student 4 mins ago
Sep 27, 2011 at 17:55	comment	added	Mariano Suárez-Álvarez		Karpilovski wrote a whole book, The Jacobson Radical of Group Algebras, about the subject... What do you mean by «simple»?
Sep 27, 2011 at 15:31	history	asked	student	CC BY-SA 3.0