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.

Filter by
Sorted by
Tagged with
0
votes
1answer
83 views

Local group rings

Let $k$ be a field of characteristic $p$ and $G$ a finite group. How do you prove that if $kG$ is local then $G$ is a $p$-group? (I know how to prove the converse but not this implication).
1
vote
1answer
62 views

Is $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n\mathbb{Z}[G]$?

Let $G$ be a group, $\mathbb{Z}[G]$ be it's group ring and $G^n$ the direct product of $n$ copies of $G$. Is the group ring $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n \mathbb{Z}[G]$? If not, is it a ...
2
votes
0answers
83 views

Units in a group ring $\mathbb{Z}[G]$

This is a homework question: For a group G, let $g\in G$ have finite order, such that $\langle g\rangle$ is not a normal subgroup of $G$. Then $\mathbb Z[G]$ has a unit other than $\pm h$ with $h\...
1
vote
1answer
675 views

A group-ring is commutative if and only if that group is abelian

Problem says: Let $R$ be a nontrivial commmutative ring and $G$ a group. Prove that $R[G]$ is commutative if and only if $G$ is abelian. I solved ($\Rightarrow $) direction as follow: Suppose ...
0
votes
0answers
89 views

Group ring generated by Z and the quaternion group

I would like to calculate general $n$th power of $i+j$ in the group ring. My idea was to find some patterns after calculating some powers of i+j, conjecture the general form of nth power of it and ...
1
vote
1answer
125 views

Natural action of $\mathbb{Z}G$ on $\mathbb{Z}$?

I'm studying projective modules and I'm having problem coming up with (or understanding) examples of non-free projective modules. I got that when a ring is a direct sum $R = A \oplus B$, both $A$ and $...
1
vote
1answer
33 views

Which element of $\mathbb{Q}C_5$ will be mapped onto $(0,4)\in \mathbb{Q} \oplus \mathbb{Q}(\zeta)$

I know that the rational group algebra of $C_5 = \langle g : g^5=1\rangle$ is $\mathbb{Q}C_5 \cong \mathbb{Q} \oplus \mathbb{Q}(\zeta)$ where $\zeta$ is a root of $x^4+x^3+x^2+x+1$. But I was ...
3
votes
1answer
103 views

Morita equivalence between $\mathbb{C}[G]$ and $\mathbb{C}[H]$?

What we can say about two groups G and H when their group rings, $\mathbb{C}[G]$ and $\mathbb{C}[H]$, are Morita equivelent?
2
votes
1answer
72 views

Structure of $\mathbb{Q}S_3$

I have an exercise to show that $\mathbb{Q}S_3 \cong \mathbb{Q} \oplus\ \mathbb{Q}\ \oplus\ M_2(\mathbb{Q}) $ , where $M_2(\mathbb{Q})$ is ring of $2$ by $2$ rational matrices and $\mathbb{Q}S_3$ is ...
0
votes
1answer
64 views

prove $\mathbb{Z}(G\times C_2)\cong (\mathbb{Z}G)C_2$

I want to prove $\mathbb{Z}(G\times C_2)\cong (\mathbb{Z}G)C_2$, where $C_2=<x| x^2=1>$, where $\mathbb{Z}(G\times C_2)$ and $(\mathbb{Z}G)C_2$ are integral group rings and I am looking for ring ...
1
vote
1answer
39 views

If a group algebra acts regularly on a module, can this module be identified as a left ideal?

To be more specific, I am looking at $F_2[D_p]$, where $D_p$ is the dihedral group of order $2p$. If this group acts regularly on the basis of a vector space $F_2^{2p}$, and there is a subspace of $...
4
votes
1answer
91 views

Is $\mathbb{Z}_p[\mathbb{Z}_p]$ a PID?

Is $\mathbb{Z}_{p}[G]$ a PID, where $G=(\mathbb{Z}_{p},+)$ is the additive group of the $p$-adics $\mathbb{Z}_{p}$? I am studying a paper where the authors implicitly use that claim, but it is ...
1
vote
1answer
56 views

Is $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID?

As the title suggests, I'm interested whether $\mathbb{Z}[\mathbb{Z}/(p)]$ a PID or not. Assume $p$ is prime. My feeling is that it is a PID, since $\mathbb{Z}/(p)$ is cyclic an morally if an ideal ...
0
votes
1answer
48 views

Constructing Factor Ring $\mathbb C / I$ where $I=_{\mathbb C[X]}\langle X^2+1 \rangle$

I'm trying to get some intuition behind the Factor Ring $\mathbb C[X] / I$ where $I=_{\mathbb C[X]}\langle X^2+1 \rangle$. 1)What does it look like in the set notation? 2)Is the equivalence ...
1
vote
0answers
44 views

Properties of Linear Transforms

Given a linear function $T: \Bbb R \rightarrow \Bbb R$ and $T(x+y) = T(x) + T(y)$, do we have that $T(x) = s(x)$ for some $s \in \Bbb R$ I'm doing an exam review for my analysis class but the ...
0
votes
1answer
108 views

Tensoring Groupring with field of fractions

Is it true that $\mathbb Z[G]\otimes_\mathbb Z \mathbb Q$ is isomorphic to $\mathbb Q[G]$ as a $\mathbb Q$-algebra? Context: If I have the Jacobian of a smooth curve $X$, call it $J_X$, then I've ...
6
votes
1answer
119 views

Prove that $u$ is a trivial unit

Let $G$ be a finite group. I want to prove that if $u\in \mathbb{Z}G$ be a torsion unit (i.e. for some $n$, $u^n=1$) of integral group ring $\mathbb{Z}G$ such that $u$ normalizes $G$, then $u$ is a ...
0
votes
1answer
189 views

Structure of induced module for subgroup $H \le G$, confused by mixing up notions of $FH$-submodule and $F$-subspace

The following is a question on a passage in the book A Course in the Theory of Groups by Derek Robinson. There he constructs the induced module $M^G$ from a $FH$-module for some subgroup $H \le G$ of ...
1
vote
1answer
70 views

Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$, where $N_G = {\sum}_{g \in G} {g} \in F[G]$. [duplicate]

I am trying to solve this Representation Theory question: Let $F$ be a field and $G$ a finite group. Let $N_G = {\sum}_{g \in G} {g} \in F[G]$. Show that the left ideal $(N_G) \subset F[G]$ is a ...
0
votes
2answers
74 views

Two ways to decompose $\mathbb Q[G]$ for $G = \{1,g,g^2\}$ and their interrelation.

Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces $$ I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \} $...
0
votes
2answers
111 views

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), \...
1
vote
1answer
49 views

Normalizer probtlem for finite nilpotent groups

Lemma- Suppose $P$ is a $p$-group contained in $G$ and $u\in N_U(G)$ where $U=U(\Bbb{Z}G)$. Then there exist $y\in G$ such that $u^{-1}gu=y^{-1}gy\ \forall\ g\in P$. We use this lemma to prove ...
1
vote
1answer
109 views

Coding theory in group rings

I am doing some extra practice problems and I got stuck on this one: Show that every element in the group ring $Z_2 C_n$ with even support (i.e. $wt(u)$ is even) is a zero-divisor. (Hint: Show that ...
2
votes
1answer
144 views

Calculating the basic algebra over a finite field in GAP

Assume $A$ is a (nonsemisimple) finite dimensional algebra over a finite field $K$ (for example a group algebra). I want to calculate the basic algebra $B$ of $A$ as a matrix algebra, constructed as ...
0
votes
1answer
53 views

twisted group ring: uniqueness of representation

$G$ is a multiplicative group, $K$ is a field. Let $\gamma$ and$\tilde{\gamma}$ be two twistings of $K^t[G]$ related by the equation $\tilde{\gamma}(x,y)=\delta(x)\delta(y)\delta(xy)^{-1}\gamma(x,y)$. ...
0
votes
1answer
139 views

How is an abelian $G$-operator group with $m1 = m$ a $\mathbb Z[G]$-module

Let $M$ and $G$ be groups. We call $M$ a $G$-group (or group with operators) if every $g \in G$ corresponds to an endomorphism of $M$, i.e. we have $$ (mn)^g = (n^g)(m^g). $$ (the application of ...
1
vote
0answers
49 views

Isomorphism map from $\Bbb{Q}(C_2\times C_2) $ to $\Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q} \oplus\Bbb{Q} $

I know how to find structure of Rational group algebras of finite cyclic groups such as for cyclic group $C_6=\langle a \rangle$ we have $\Bbb{Q}C_6 \cong \Bbb{Q} \oplus \Bbb{Q}\ \oplus\ \Bbb{Q}(\...
8
votes
1answer
890 views

Compute the Jacobson radical of the group ring $\mathbb{F}_2S_3$.

Compute the Jacobson radical and the maximal semisimple quotient of the group ring $\mathbb{F}_2S_3$ of the symmetric group on three letters over the field with two elements, and compute the same ...
0
votes
1answer
24 views

Show that $[u,g]\in G'$ where $u\in N_U(G)$

Let $G=\langle H,g\rangle$ where H is an abelian subgroup of index $2$. Let $\Bbb{Z}G$ be the group ring and $u$ be a unit of $\Bbb{Z}G$ which normalizes $G$. Then we can write $u$ as $\alpha_1+\...
1
vote
1answer
27 views

order of $h_0 $ divides augmentation of $\alpha\in \Bbb{Z}H $

Let $H$ be an abelian group and $\Bbb{Z}H$ be its integral group ring. Now let $\alpha=\sum_{h\in H}a_h.h\in \Bbb{Z}H$ and $\alpha(1-h_0)=0$ for some $h_0\in H$. Why does this imply that order of $...
2
votes
2answers
42 views

Why does $\Bbb{Z}(G/H)$ has only trivial units?

Let $H$ be an abelian subgroup of index $2$ in $G$, and $G=\langle H,g \rangle$ then why is it only units of $\Bbb{Z}(G/H)$ are trivial (i.e. of the form $\pm\bar{g}$)? An arbitrary element of $\Bbb{...
1
vote
0answers
129 views

Jacobson radical of integral group ring ZS3

I want to know whether Jacobson radical of integral group ring ZS3 (S3 is symmetric group) is known or not. Please help me. Tell me about any reference. Also if anyone know about the maximal ideals ...
3
votes
1answer
257 views

units of group ring $\mathbb{Q}(G)$ when $G$ is infinite and cyclic

How would I be able to describe all units of the group ring $\mathbb{Q}(G)$ where $G$ is specifically an infinite cyclic group?
9
votes
1answer
947 views

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( \...
0
votes
1answer
207 views

group ring Z(G) is isomorphic to tensor algebra T(G)

Well, I've been learning Tensor algebra and related topics. So, I wonder tensor algebra is related to somehow familiar with me, so called group ring. $\mathbb{Z}$: a ring of integers. Let G be a ...
1
vote
1answer
28 views

Equivalent condition for normalizer problem

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 ...
1
vote
1answer
157 views

Difference between $\mathbb{Z} G$-module and $G$-module

I am studying Group Cohomology, but I am a little confused about $\mathbb{Z} G$-module and $G$-module. Some text uses the $G$-module for group cohomology, but I thought group cohomology of $G$ is ...
2
votes
1answer
43 views

How do I view $k$ as $kG$-module?

How could I view $k$ as $kG$-module? Does $kG$ acts on $k$ by $rg\cdot r'=rr'$, just ignore $g$? Could we do that?
2
votes
1answer
90 views

Conditions for free/projective/flat module over a groupring

Let $H \subset G$ be a subgroup of the group $G$. When is $\mathbb{Z}[G]$ a free/projective/flat $\mathbb{Z}[H]$-module? If $\mathbb{Z}[G]$ is a free $\mathbb{Z}[H]$-module then there is a $n \in \...
2
votes
1answer
44 views

If $H\leq G$ and $|H|$ is invertible in a ring $R$, why does $RG/H\simeq e_HRG$?

Suppose $R$ is a commutative unital ring, $G$ a finite group, and $H$ a subgroup of $G$ whose order is invertible in $R$. Defining $e_H=|H|^{-1}\sum_{h\in H} h$, why is $RG/H\simeq e_HRG$? This ...
1
vote
0answers
33 views

What is the augmentation ideal of $\Bbb{Z}S_3$

I know that $\Delta_{\Bbb{Z}}(G)$ is the $\Bbb{Z}-$ module generated by elements of form $\{g-1\ \forall\ g\in G\}$. But how do we find them or what it looks like. I was thinking about finding aug ...
2
votes
0answers
28 views

Express $[RG,RG]$ as $R-$ linear span of $[g,h]$

Let $R$ be a ring and $G$ be a group and $RG$ be the group ring. Denote by $[R,R]$, the additive subgroup generated by all lie products $[x,y]=xy-yx , \forall\ x,y\in R$. Then how is this that $[\...
6
votes
2answers
180 views

Is it true that $(R\times S)[G]\cong R[G]\times S[G]$?

I know for two groups $G, H$ (not necessarily finite) we have $R[G\times H]\cong (R[G])[H]$, but I was wondering if we had a similar statement for rings $R,\,S$. In other words, if $R,\,S$ are two (...
2
votes
2answers
134 views

Projective but not free module over groupring

Let $G$ be a nontrivial finite group and consider the groupring $\Bbb QG$. My question is whether we can find a module over $\Bbb QG$ that is projective but not free?
1
vote
1answer
105 views

Show $D = \delta_1 +\cdots+ \delta_m$ is in the center of the group ring $RG$ .

Let $\Delta$ = {$ \delta_1,\dots, \delta_m$} be a conjugacy class in a finite group $G$. Prove that the element $D = \delta_1 + \cdots + \delta_m$ is in the center of the group ring $RG$ . Hint: ...
1
vote
0answers
96 views

Calculation in a Group Ring

I have some problems with the verification of the third equation in Lemma 1 in this paper. First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ ...
1
vote
1answer
289 views

Decomposition of group algebra $\mathbb{C}[G]$ [duplicate]

Let $G$ be a finite group. By Wedderburn theorem, we see that $\mathbb{C}[G]$ is product of matrix algebras. "Linear Representations of Finite Groups" by Serre has an explanation for this result ...
4
votes
2answers
83 views

Is this module over this group algebra projective?

Assume that $G$ is a finite group. Let $k$ be a field. Let $\varepsilon$ be the augmentation $kG\rightarrow k$. Consider the following map $\varepsilon\otimes id:k[G]\otimes_k k[G]\rightarrow k[G]$...
0
votes
0answers
38 views

Elements that aren't left zero divisors are invertible for certain group algebra

Let $G$ be a finite group and $F$ a finite field with co-prime orders. Show that in the group algebra $F[G]$, if $x \in F[G]$ is not a left zero divisor then it is invertible. Thoughts so far: By ...
1
vote
1answer
145 views

Is the inclusion map always a module homomorphism?

Suppose $R$ is some commutative ring and $G$ a finite group so that $R[G]$ is the usual group ring. If $M$ is some $R[G]$-module, then we can inject $M\hookrightarrow M\oplus R[G]^n$ for some $n\geq 1$...