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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.

The group ring $R[G]$ is constructed in the following way. The set $R[G]$ is the free $R$-module on the elements of $G$, equipped with the multiplication given by the operation in $G$ extended distributively to all elements in the free module.

A special case of this construction is group algebra, which occurs naturally in representation theory. It turns out that every group representation $\rho:G\rightarrow GL(V)$ corresponds to an $R[G]$ module structure on $V$. This connection ties the representation theory of groups to the module theory of group algebras.

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How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation theory of $G$ over $K$, as for instance if…
Will
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What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?

The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then $k[G]$ is a direct sum of matrix algebras, one for…
Qiaochu Yuan
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When is a group ring an integral domain

If $R$ is an integral domain (I am having $\mathbb{Z}$ or a field in mind) and $G$ a (not necessarily finite) group then we can form the group ring $R(G)$. Note that if $g^{n+1} = e$ then $(e-g)(e+g\ldots + g^n) = e - g^{n+1} = 0$. This means if…
mna
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Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-module, and let $P_3\to P_2\to P_1\to…
Thomas Browning
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Why is the constant term of $(1+x+y+xy)^n$ equal to $\frac{1}{2}\binom{2n}{n}$？

If we define this: for any $x,y$ such that $x^2=y^2=1,xy\neq yx$, express in terms of $n$ the constant term of the expression $$f_{n}=(1+x+y+xy)^n\,.$$ I guess this result is $\dfrac{1}{2}\binom{2n}{n}$. for $n=1$, we have $f_{1}=1+x+y+xy$ the…
math110
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The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite group and $k$ a field of characteristic $p>0$. The…
Benjamin Steinberg
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Augmentation ideal of the group ring

Let $G$ be a group and $I_G$ be the augmentation ideal of the group ring $\mathbb{Z}G$, i.e. $I_G$ consists of formal linear combinations $\sum n_i g_i$ ($n_i\in\mathbb{Z}$, $g_i\in G$) such that $\sum n_i=0$. Is there a characterization of the…
user8268
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Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ is a nilpotent ideal ($p$ is a prime, $G$ is a $p$-group)

Let $p$ be a prime and let $G$ be a finite group of order a power of $p$ (i.e., a $p$-group). Prove that the augmentation ideal in the group ring $\mathbb{Z}/p\mathbb{Z}G$ (to be read as $\left( \mathbb{Z}/p\mathbb{Z} \right) G$) is a nilpotent…
Liza
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Minimal counterexamples of the isomorphism problem for integral group rings

The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been done towards solving this and related problems. See…
Martin Brandenburg
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Example of non isomorphic groups with isomorphic group algebras

Below is the construction of two non isomorphic groups, $G_1$ and $G_2$ such that $KG_1 \cong KG_2$ for any field $K$. (My Doubts lie within.) Consider two groups $Q_1=\langle x_1,y_1,z_1\ |\ x_1^5=y_1^5=z_1^5=1, [x_1,y_1]=z_1, z_1\ \text{is…
Bhaskar Vashishth
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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 make a ascending chain of ideals that won't stop,…
kpax
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How to recover the integral group ring?

I would like to solve the following exercise: Suppose $R$ is a commutative semisimple ring of characteristic $p^t, t\geq1$, and we have two finite groups $G_1=H_1 \times A_1$ and $G_2=H_2 \times A_2$. Now $H_1$ and $H_2$ are finite $p$-groups,…
Boris Datsik
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Isomorphism between $I_G/I_G^2$ and $G/G'$

Ok, this has been bugging me for a while, and I'm sure there's something obvious I'm missing. The references I've looked at for this result in an effort to resolve the issue didn't address it. $G$ is a group, $\mathbb{Z}[G]$ its integral group ring,…
Zev Chonoles
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Which rings arise as a group ring?

Let $R$ be an arbitrary associative ring with identity. When does there exist a group $G$ and a field $F$ such that $F[G] = R$? Do we obtain more rings as $F[G]$ if we loosen the condition that $F$ be a field? When can we get $G$ to be a finite…
Cloudscape
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Do group rings appear outside of representation theory?

I am particularly concerned with finite groups. I have seen group rings used in the fundamentals of representation theory as the dual notion to representations. I haven't ever seen them anywhere else. Are there problems in (or applications of)…
Samuel Handwich
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