Consider group algebra k[G] of finite group G.
If k is alg.closed then every irrep lives there with multiplicity equal to dimension.
(More conceptually as bimodule over GxG it is multiplicity free and all irreps live there).


Now let k=F\_p what is known then ?
Is true that all irreps of G live in k[G] ? (Seems obviously yes... but to be sure)
But multiplicities ?


If assume order of G does not divide p ? 




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Example:


G=Z/nZ, the F\_p[G] = F\_p[x] / (x^p-1) and every irreducible polynom generates ideal and hence subrepresentation, so these are quests about irred. polynoms. E.g. like this:


<https://math.stackexchange.com/questions/172534/for-which-values-of-n-is-the-polynomial-px-1xx2-cdotsxn-irreducible/172540#comment396429_172540>


<https://math.stackexchange.com/questions/172468/for-what-n-k-there-exists-a-polynomial-px-in-f-2x-s-t-degp-k-and>




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Motivation: ["cyclic error correcting codes"](http://en.wikipedia.org/wiki/Cyclic_code) are exactly the ideals in F\_p[x] / (x^p-1) = 
 F\_p[G], why take G= Z/nZ ? may be take other groups give "better" codes ? 



Structure of F_p[G], for finite group G ?  
It seems natural to work with an algebraically closed field of characteristic $p$, or, less restrictively, a splitting field of characteristic $p$ for $G$. For example, any field containing the primitive $m$-th roots of unity, where $|G| = p^{a}m$ and $p$ dos not divide $m$, so I assume now that $k$ is algebraically closed of characteristic $p.$ We have entered the realm of modular representation theory, whose theory was first extensively developed by Richard Brauer. By now the basic theory is covered in numerous texts (eg by Alperin, by Curtis-Reiner).


The indecomposable direct summands of the group algebra $kG$ are the so-called projective indecomposable modules (sometimes called principal indecomposables). The number of isomorphism types of these is the number of conjugacy classes of elements of order prime to $p$ of $G.$ Each of these has a simple socle (the socle of module is its largset semisimple submodule) and a unique maximal submodule (so a simple head, or largest semisimple quotient module). Because the group algebra is a symmetric algebra, the socle and head of $P$ are isomorphic. If $P$ is one of these projective indecomposable modules, and has (simple) socle $S$, then up to isomorphism, $P$ occurs ${\rm dim}\_{k}(S)$ times as a summand of the group algebra $k[G].$ Hence $S$ occurs $dim\_{k}(S)$ times (up to isomophism) as a summand of the socle of the regular module $kG.$ Also, $S$ occurs ${\rm dim}{k}(P)$ times as a composition factor of the regular module $k[G].$ Every simple module for the group algebra $k[G]$ has a projective cover, and these all occur as direct summands of the group algebra. As Mariano remarks in his comment, the theory of non-projective indecomposable modules is much less transparent. The results over non-splitting field can be recovered using Galois theory and Clifford theory. However, modular representation theory is much richer and diverse than this brief description allows for. These basic facts (and much much more) were all known to Brauer, perhaps sometimes in different language, and these are just the beginnings. If $|G|$ is not divisible by $p,$ the projective indecomposable $P$ and its socle $S$ are the same module, and the theory degenerates to the semisimple case, which is much like the characterisic zero situation. 


