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act by conjugation on the [[Group algebra|group algebra]] $ kG $ with fixed points $ ( kG ) ^ {G} = Z ( kG ) $.

...in the group algebra is induced by the group multiplication. The algebra $KG$ is isomorphic to the algebra of functions defined on $G$ with values in $K ...lows: If $G$ is a finite group and $K$ is a field, then the group algebra $KG$ is semi-simple if and only if the order of $G$ is not divisible by the cha

...bra $H$ (cf. [[Hopf algebra|Hopf algebra]]) satisfying $H\otimes_R K \cong KG$ as $K$-Hopf algebras. ...i(g)}$ be an element in $R$ of value $\xi(g)$. Then the $R$-Hopf order in $KG$ determined by $\xi$ (called a Larson order) is of the form

$$\lim_{n\to\infty}\frac{\sum_{k=0}^nT^kf(w)}{\sum_{k=0}^nT^kg(w)}$$ ...tly large $n$ differs from zero, i.e. where at least one of the numbers $T^kg(w)>0$.

or, equivalently, a module action of the [[Group algebra|group algebra]] $ KG $ is the sum of the characters of the quotient $ KG $-

...et of representatives $\{\bar{g}\}$ of $G/N$ in $G$. Then every $\alpha\in KG$, the [[group algebra]] of $G$, can be written as a unique sum $\alpha=\sum $$ KG=KN(G/N,\rho,\sigma) . $$

...ield and $G$ is a finite extension of a polycyclic group, then any simple $kG$-module is finite-dimensional over $k$. In any group, the product of two lo

M - kF &N - kG \\

...V$ is a linear vector space over a field $K$, then the specification of a $KG$-module structure on $V$ is equivalent to giving a representation of $G$ in ...e. For example, if $M$ is a finitely-generated module over the group ring $KG$ of a finite group $G$, where $K$ is a field of characteristic coprime with

...(or $s$). For this one has to find the remainders from the division of $x^kg(x)$ by $f(x)$, $k=0,\cdots,n-1$. Let these be

linearity to the mapping $ \chi : {KG } \rightarrow K $).

is called an Einstein metric if $ \mathop{\rm Ric} _ {g} = kg $

\mathrm{Ext} _ {kG} ^ {n} ( k, A),

...nt(K) \to \Cent({}^gK)$ the map given by $(\inn g)_K(\psi)(k) = \psi(g^{-1}kg)$ for all $k \in {}^gK$. We will also denote the character $(\inn g)_K(\psi