# Questions tagged [group-rings]

A group ring $R[G]$ is a ring constructed from a group $G$ and ring $R$. A special case of this construction is group algebra, which occurs naturally in representation theory.

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### Is there an 'easy' way to calculate $K_0(\mathbb{Z}[C_p])$?

For $C_2$ the cyclic group of order 2, I want to calculate $\tilde{K}_{0}(\mathbb{Z}[C_2])$. Now so far, I know by a theorem of Rim that $\tilde{K}_{0}(\mathbb{Z}[C_2])$ is isomorphic to the ideal ...
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### If $k$ is a field of characteristic $p>0$ and $G$ is a finite group of order divisible by $p$ then $k[G]$ is not a semi-simple ring.

If $k$ is a field of characteristic $p>0$ and $G$ is a finite group of order divisible by $p$ then $k[G]$ is not a semi-simple ring. My failed attempt: Since $G$ is finite and $p$ divisdes $|G|$ ...
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### Canonical homomorphism from a group ring to its subring.

If $R(G)$ is a group ring constructed from a group $G$ and a ring $R$, and $H\leq R(G)$ is a subring, then must there be a surjective homomorphism from $R(G)$ to $H$ in general? What if $G$ is a ...
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### Best book to understand representation theory.

I have tried to read representation and character theory from a few books but none of them was working for me, like Lieback and Serre. I want to understand representation and character to use them and ...
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### What is $\mathbb{Q}[\mathbf{Z}/n\mathbf{Z}]$?

I was thinking, you can send $\sum f(g)g\in\mathbb{Q}[\mathbb{Z}/n\mathbb{Z}]$ to $(f(g))_{g\in G}\in \mathbb{Q}\times\cdots\times\mathbb{Q}$. This is surjective and has zero kernel, so this map ...
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### If $G$ is an infinite group, then the group ring $R(G)$ is not semisimple.

Let $R$ be a ring and $G$ an infinite group. Prove that $R(G)$ (group ring) is not semisimple. My idea was to suppose it is semisimple, then $R(G)$ is left artinian and $J(R(G))=0$. I was trying to ...
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### If $G \cong \mathbb Z/3\mathbb Z$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$
Let $G = \{1,g,g^2\}$ be the cyclic group of order three. Consider the group ring $\mathbb R[G]$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$ with the isomorphism  \varphi(1) = (1,0), \...
Let $\cal{U}$ be the unit group of group ring $\Bbb{Z}G$ then the Normalizer Problem (NP) states that $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}=\cal{Z(U)}$. Now why (NP) is equivalent to saying that ...