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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Can the integral group ring construction be extended to groupoids in such a way that it provides a functor from Groupoids to Rings?

If $G$ is a group, and $R$ is a ring, one can construct the group ring $R[G]$, which is a free $R$-module with basis $G$, and with multiplication coming from the original multiplication of $G$. http:/...
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198 views

Rational group algebras and maximal orders

Let $G$ be a finite group, and $\mathbb{Q}[G]$ be the rational group algebra. Then the group ring $\mathbb{Z}[G]$ is an order in $\mathbb{Q}[G]$, but is not in general a maximal order. What can we ...
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271 views

Jacobson radical of the integral group ring

I am trying to prove that the Jacobson radical of the integral group ring $\mathbb{Z}G$ for a finite group is zero. Most of what I find on semisimplicity, Jacobson semisimplicity, has to do with ...
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144 views

Is the Transpose of a Representation an Equivalent Representation?

Suppose we are working over $\mathbf{Z}[G]$ where $G$ is finite. Suppose further we have two representations $\rho$ and $\rho^\prime$ such that $\rho^\prime=(\rho)^T$. Can we say that these two ...
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190 views

Consider $\mathbf{Z}G$, $G$ finite. If the characters of two $\mathbf{Z}G$-modules are equal, does it follow that the modules are isomorphic?

So I have recently started to delve into integral representation theory and I was wondering if a particularly useful theorem survives the transition to integral rep theory. Basically, suppose we have ...
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77 views

Identities of a group ring

Let $R$ be a ring with identity $1_R\neq 0$, and let $G$ be a finite group with group operation written multiplicatively. Then we can form the group ring $$RG = \{a_1g_1+a_2g_2+\cdots+a_ng_n\ |\ a_i\...
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139 views

Is the coefficient ring $R$ of a group ring $RG$ necessarily projective as an $RG$-module?

So this may be a trivial question but I am new to the idea of group rings. Suppose we have a ring $R$ and a group $G$, I was wondering if the trivial $RG$-module $R$ is projective? In which case, how ...
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47 views

Embedding the base ring in the augmentation ideal of a group algebra

Let $G$ be a finite group. Then the group algebra $\mathbb{Q}G$ trivially contains $\mathbb{Q}$. But when (i.e. for which $G$) does the augmentation ideal $I_G=\{\sum_{g\in G} r_g\,g \mid \sum_{g\in ...
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436 views

Projective modules over $kG$ equivalent to injective.

Let $k$ be a field and $G$ is finite group. I want to prove that a $kG$ module $P$ is projective iff it's injective. I proved that if module is projective then it's injective. 1) $kG$ is injective ...
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76 views

group algebra and kernel

I want to prove this lemma Let $G$ be a group, $H$ an invariant subgroup, $k$ a field; consider the canonical map $\varphi \colon k[G] \to k[G/H]$; then the kernel of $\varphi$ is the ideal ...
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124 views

Are augmented algebra maps of group algebras group homomorphisms?

Given a finite group $G$ and a field $k$, we can form the group algebra $kG$ with basis the elements of $G$. There is a natural augmentation $\varepsilon\colon G\to k$ that sends an element to the sum ...
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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 ...
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137 views

A question about Group Rings

Let $R:=\mathbb{Z}_p[C_{p^\infty}]$ be the group ring of a Prufer group over the field of integers module a prime $p$. We have $C_{p^\infty}=\langle u_1, u_2, ..., u_n, ... |\,\,\,\, u_1^p=1,\,\,u_{...
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247 views

Are there homomorphisms of group algebras that don't come from a group homomorphism?

Given a finite group $G$, one can define the group algebra $\mathbb{C}[G]$ as the algebra having the elements of $G$ as a basis, with the multiplication of $G$. Clearly, any group homomorphism induces ...
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$\mathbb{C}[x,y\,|\,x^m=1,y^n=1]\cong\mathbb{C}[z\,|\,z^{mn}=1]$ as complex algebras?

If $G$ is any finite abelian group and $K$ an algebraically closed field with $|G|\neq 0$ in $K$, then the group algebra $K[G]\cong M_{n_1}(K)\times\cdots\times M_{n_k}(K)$ by Maschke's and Artin-...
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110 views

Existence of nontrivial unit in $\mathbb{Q}[G]$, where $G$ is finite.

Suppose $G$ is a finite group of order $|G|>1$, and $\mathbb{Q}[G]$ is the group ring. I'm curious about an example of a nontrivial invertible element, i.e., one that is not of the form $ag$, with $...
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137 views

An invariant submodule of a projective module

Let $R$ be a commutative unital ring and $S$ be a normal subgroup of a finite group $G$. If $P$ is a finitely generated projective left $R[G]$-module then is the submodule of $S$-invariants $$P^S:= \{...
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434 views

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 ...
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770 views

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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Modules over a group algebra

Sorry for the bad quality picture. The minuses should of course be equals. For the first, I'm thinking Schurs Lemma will be involved. For ci) I need to show it's a subspace and is closaed under the ...
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81 views

Elements of Group Rings, Are they Reducible?

In general, elements of group rings are written as sums of the group elements multiplied by scalars from the ring, correct? What is the utility of that, if we don't know how to reduce the elements of ...
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276 views

Units in finite polynomial rings

Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where $...
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123 views

Conjecture about some group ring representations.

In this link : http://bandtechnology.com/PolySigned/ A set of numbers is described : $P(N)$. $ P(3),P(4),P(5),... $ are all (algebraicly closed) group rings. Identify $PN$ with $R[X_N]/(1+X_N+(X_N)^...
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168 views

Showing the augmentation ideal of $\mathbb{F}[G]$ is a maximal ideal

Given a basis of $\Bbb R^n,\ G:=\{e_0,...,e_{n-1}\}$, we define multiplication on the elements of the basis by $e_i\cdot e_j=e_{i+j}$ (where $i+j$ is calculated modulo $n$). For a field $\Bbb F$ we ...
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2answers
1k views

Units of a group ring.

Let $\mathbb{Q}$ be the rationals and $G$ a group. Then we consider the group ring $\mathbb{Q}[G]$. Since the operation on $\mathbb{Q}[G]$ restricted to $G$ is just the group operation, I know that ...
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239 views

Isomorphism of Tensor Product over a Group Ring.

Let $\mathbb{Q}$ be the rationals and $\mathbb{Z}$ integers. Let further $p$ be prime and $t\in \mathbb{Z}$ such that $p \mid t$. Then $\mathbb{Z}_{(p)}$ is the local ring. Let $G < H$ be groups, ...
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90 views

Calculation on Infinite Groups and Group Rings.

Let $G$ be an infinite group and $x \in R[G]$ the group ring of $G$ over $R$. If $gx-x=0$ $\forall g \in G, g\neq 1$, then follows $x=0$. I need a nice proof. I already have one dealing with the '...
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177 views

Zero divisor conjecture

Let $K$ be a field and $G$ be a group. Then $K[G]$ is a domain iff $G$ is torsion-free. I know that "$\Leftarrow$" is conjectured to be always true. But what about the other direction?
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Finite $\Delta$-module of $p$-power order

I have a question concerning lemma, that I want to prove: Let $p$ be a prime and $\Delta$ be a finite group of order prime to $p$. Let $M$ be a finite $\Delta$-module of order a power of $p$. Then ...
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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\...
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268 views

Generation of left ideals in group rings

This looks like an elementary exercise on group rings (I heard it somewhere), nonetheless it seems to be non-trivial to me. Any references much appreciated. Suppose that we are given an (infinite) ...
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416 views

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 ...
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find a special element in group algebra

Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual ...
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178 views

An ideal generated by a nontrivial idempotent is not a free module

Let $R = \Bbb QG$ with $G$ isomorphic to a cyclic group of order $2$, and $x$ its generator. I'm trying to show that $$\frac{1}{2}(1+x)R$$ is not a free $R$-module. I found out that $\frac{1}{...
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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]...
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For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?

I'm reading the Atiyah-MacDonald book on Commutative Algebra. At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is: $G$ = finite group, $...
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115 views

Prove an inequality in a group ring

Let $$G=\bigoplus_{n\in\mathbb{Z}}\left(\mathbb{Z}/2\mathbb{Z}\right)_n$$ be a group, and for any $n\in \mathbb{Z}$, denote $\delta_n$ to be the element in $G$ with $n$-th coordinate $1$ and zero at ...
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262 views

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 ...
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Artin-Wedderburn decomposition of a particular group ring

I am trying to do a question from an algebra qualifying exam: Decompose the group ring $\mathbb{F}_5[S_3]$ as a product of simple rings. By Maschke's theorem since $\mathrm{char}(\mathbb{F}_5) \...
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278 views

How to turn a group ring $R(G)$ into a ring?

Let $R(G)$ be a given abelian group ring. Any abelian group ring is isomorphic to an abelian ring. I know how to express (isomorphism) some group rings as a ring. But I wonder if there is a general ...
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696 views

Quotient of ideal in group ring is isomorphic to abelianization [duplicate]

Let $G$ be a group and $\mathbb Z G$ the group ring over the integers. Let $I$ be the ideal of elements $\sum_{g\in G} n_g g$ with $\sum_{g\in G} n_g = 0$. I am trying to prove that $I/I^2$ is ...
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140 views

$G/G' \cong I/I^2$ where $I$ is the augmentation ideal [duplicate]

Possible Duplicate: Isomorphism between $I_G/I_G^2$ and $G/G'$ Let $G$ be a finite group. Let $I\unlhd\mathbb{Z}[G]$ be the augmentation ideal of the integral group ring $\mathbb{Z}[G]$. I'm ...
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An algebraic algorithm for finding inverses in the group algebra

This is an extension to my earlier question. Is there a purely algebraic algorithm to find inverses in the group algebra? For example, in the group algebra $\mathbb{C}S_{4}$, how would one go about ...
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1answer
768 views

How to calculate inverses in the group algebra

Is there an algorithm to calculate the inverse of an element in the group algebra? For example, does the element $(1 2 3) + 2 . (1 2)(3 4)$ in the group algebra $\mathbb{C} S_{4}$ have an inverse, ...
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Finding elements in a group ring

Suppose we have a discrete group $G$, finite or infinite, on which we form the group algebra $\mathbb{F}_2[G]$. Suppose also that we have a map $S$: $$S : \mathbb{F}_2[G] \to \mathbb{F}_2[G]: \sum{...
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1k views

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 $\...
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3answers
2k views

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 ...
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386 views

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 ...