Group algebras and their modules

Is the algebra of quaternions a real group algebra:

Let $V$ be a vector space over a field $F$ with basis $\left(e_{1}, e_{2}, e_{3}\right)$, $\varphi : F\left[\mathbf{S}_{3}\right] \rightarrow \operatorname {End} V$ the homomorphism where $\varphi (\sigma )\left(e_{i}\right)=e_{\sigma (i)}$ for all $\sigma \in \mathbf{S}_{3}(i=1,2,3)$. Find the dimension of the kernel and the dimension of the image of the homomorphism $\varphi$.

Find a basis for the kernel of the homomorphism $\varphi : \mathbb {C}\left(\langle a\rangle_{n}\right) \rightarrow \mathbb {C}$ given by $\varphi (a)=\varepsilon$, where $\varepsilon$ is an $n$-th root of 1.

Let a group $H$ be isomorphic to a quotient group of a group $G$. Prove that $F[H]$ is isomorphic to a quotient algebra of the algebra $F[G]$.

Let $G=G_{1} \times G_{2}$. Prove that

Let $G$ be a finite group, $R$ the set of maps from $G$ to a field $F$. Define operations on $R$ by setting, for $f_{1}, f_{2} \in R$,

Prove that $R$ is an algebra over the field $F$ and that the map

from $R$ to $F[G]$ is an isomorphism of algebras.

Prove that if a group $G$ contains elements of finite order, then the group algebra $F[G]$ has zero divisors.

Prove that every irreducible $F[G]$-module is isomorphic to a quotient module of the regular $F[G]$-module.

Find all commutative two-sided ideals of the group algebra $\mathbb {C}\left[G\right]$ for:

Find all elements $x$ of the group algebra $F[G]$ satisfying $x g=x$ for every $g \in G$.

Find a basis for the center of the group algebra of the groups:

Prove that the group algebra $A$ of a free abelian group of rank $r$ has no zero divisors. The field of fractions of $A$ is isomorphic to the field of rational fractions in $r$ variables.

Let $A$ be a ring, $V$ an $A$-module, and $V=U \oplus W$, where $U$ is an irreducible module and $W$ contains no submodules isomorphic to $U$. Prove that if $\alpha$ is an automorphism of the module $V$, then $\alpha (U)=U$.

Let $A$ be a ring, and suppose an $A$-module $V$ decomposes as a direct sum of submodules $V=U \oplus W$, with $\varphi : U \rightarrow W$ a homomorphism of $A$-modules. Prove that $U_{1}=\left\{ x+\varphi (x) \mid x \in U\right\}$ is an $A$-submodule of $V$ isomorphic to $U$, and $V=U_{1} \oplus W$.

Let $A$ be a finite-dimensional semisimple algebra over $\mathbb {C}$ and let an $A$-module $V$ decompose as a direct sum of pairwise nonisomorphic irreducible $A$-modules: $V=V_{1} \oplus \ldots \oplus V_{k}$. Find the automorphism group of the module $V$.

Let $A$ be a finite-dimensional semisimple algebra over $\mathbb {C}$ and let an $A$-module $V$ be the direct sum of two isomorphic irreducible $A$-modules. Prove that the automorphism group of the $A$-module $V$ is isomorphic to $\mathbf{G} \mathbf{L}_{2}(\mathbb {C})$.

Let $A$ be a finite-dimensional semisimple algebra over $\mathbb {C}$ and $V$ an $A$-module finite-dimensional over $\mathbb {C}$. Prove that $V$ has finitely many $A$-submodules if and only if it is a direct sum of pairwise nonisomorphic irreducible $A$-modules.

Let $G$ be a finite group, $F$ a field of characteristic 0, and let the group algebra $A=F[G]$ be regarded as a left module over itself. Prove that for any submodule $U$ of it and homomorphism of $A$-modules $\varphi : U \rightarrow A$, there exists an element $a \in A$ such that $\varphi (u)=u a$ for all $u \in U$.

For which finite groups is the complex group algebra simple?

Let $A=F[G]$ ($F$ a field), $G$ a finite group of order $n>1$, and, for $n \cdot 1 \neq 0$, set

Prove that $A e_{1}$ and $A e_{2}$ are proper two-sided ideals and $A=A e_{1} \oplus A e_{2}$.

Prove that the equality

in the group algebra $F[G]$ defines on the space $F[G]$ a symmetric bilinear function, and that the kernel of this function $f$ is a two-sided ideal of $F[G]$.

Let $G$ be a finite group, $f$ the bilinear function on $\mathbb {R}\left[G\right]$ defined in problem 71.21. Prove that $f$ is nondegenerate, and find the signature of the function $f$ for the groups:

Let $H$ be a subgroup of a group $G$, and let $\omega (H)$ be the left ideal of $F[G]$ minimal among the left ideals containing $\left\{ h-1 \mid h \in H\right\}$. Prove that if $H$ is a normal subgroup, then the ideal $\omega (H)$ is two-sided.

Decompose into a direct sum of fields the group algebras of the group $\langle a\rangle_{3}$ over the fields of real and complex numbers.

Prove that $\mathbb {Q}\left[\langle a\rangle_{p}\right]$ ($p$ a prime number) is a direct sum of two two-sided ideals, one of which is isomorphic to $\mathbb {Q}$, and the other to $\mathbb {Q}(\varepsilon )$, where $\varepsilon$ is a primitive $p$-th root of 1.

Let $G$ be a finite group, $\operatorname {char} F$ not dividing $\left|G\right|$, $I$ an ideal of $F[G]$. Prove that $I^{2}=I$.

Find the idempotents and minimal ideals in the rings:

Let $G$ be a finite group. Prove that for any $a \in \mathbb {C}\left[G\right]$, the equation $a=a x a$ is solvable in $\mathbb {C}\left[G\right]$.

How many distinct two-sided ideals are there in the algebra:

For which finite groups $G$ is the group algebra $\mathbb {C}\left[G\right]$ a direct sum of $n$ matrix algebras, if:

Let $G$ be a group, $A$ an algebra with unity over a field $F$, $\varphi$ a homomorphism $G \rightarrow A^{*}$. Prove that there exists a unique homomorphism $F[G] \rightarrow A$ whose restriction to $G$ coincides with $\varphi$.

Prove that if char $F$ does not divide the order of a finite group $G$, then any two-sided ideal of the group algebra $F[G]$ is a ring with unity. Is this statement true for arbitrary algebras with unity?

Let $F$ be a field of characteristic $p>0$, $p$ dividing the order of a finite group $G$, and

Prove that $F[G] u$ is a submodule of the left regular module that is not a direct summand.

Let $G=\langle a\rangle_{p}$, $F$ a field of characteristic $p$, $\Phi : G \rightarrow \mathbf{G L}_{2}(F)$, where

is a representation of the group $G$. Indicate an $F[G]$-submodule $U$ of the regular representation $V=F[G]$ such that the representation of $G$ on $V / U$ is isomorphic to $\Phi$. For which $p$ is the representation $\Phi$ isomorphic to the regular representation?

Prove that the algebra $\mathbb {F}_{2}\left[\langle a\rangle_{2}\right]$ is not a direct sum of minimal left ideals.

Let $H$ be a $p$-group that is a normal subgroup of a finite group $G$, $F$ a field of characteristic $p$.

Prove that all ideals of the group algebra of an infinite cyclic group are principal.

Prove that a cyclic module over the algebra $F\left[\langle a\rangle_{\infty }\right]$ is either finite-dimensional over $F$, or isomorphic to the left regular $F\left[\langle a\rangle_{\infty }\right]$-module.

Let $A=\mathbb {C}\left[\langle g\rangle_{\infty }\right]$, $P=A x_{1} \oplus A x_{2}$ a free $A$-module with basis $\left(x_{1}, x_{2}\right)$, $H$ the submodule of $P$ generated by the elements $h_{1}, h_{2}$. Decompose $P / H$ into a direct sum of cyclic $A$-modules and find their dimensions, if:

Let $\mathcal{A}, \mathcal{B}$ be linear operators on $V=F[x]$, $\mathcal{A}(f(x))=f^{\prime }(x)$, $\mathcal{B}(f(x))=x f(x)$. Prove that the map $\varphi : g \mapsto \mathcal{A B}$ extends to a homomorphism $F\left[\langle g\rangle_{\infty }\right] \rightarrow \operatorname {End} V$, and find $\operatorname {Ker} \varphi$.

Let $M$ be a maximal ideal of the algebra $A=F\left[\langle a\rangle_{\infty }\right]$ and $r=\operatorname {dim}_{F}(A / M)$.

Prove that:

Prove that the group algebra of a free abelian group of finite rank is Noetherian.

Prove that in the group algebra of a free abelian group of finite rank, the theorem on the existence and uniqueness of factorization into prime factors holds.

Factor into a product of prime elements the following elements of the group algebra $A=\mathbb {C}\left[G\right]$ of a free abelian group $G$ with basis $\left(g_{1}, g_{2}\right)$:

Let $G$ be a free abelian group with basis $\left(g_{1}, g_{2}\right)$. Find the quotient algebra of the group algebra $A=F[G]$ by the ideal $I$ generated by the elements:

Prove that if a group $G$ is finite and the algebra $\mathbb {C}\left[G\right]$ has no nilpotent elements, then $G$ is commutative.

Let $H$ be a normal subgroup of a group $G$, $V$ some $F[G]$-module, and $(H-1) V$ the linear span of elements of the form $(h-1) v$, where $h \in H, v \in V$.

Prove that:

Prove that the complex group algebras of the groups $\mathbf{D}_{4}$ and $\mathbf{Q}_{8}$ are isomorphic.

Find the number of pairwise nonisomorphic complex group algebras of dimension 12.

Prove that the number of summands in the decomposition of the group algebra of the symmetric group $\mathbf{S}_{n}$ over the field $\mathbb {C}$ into a direct sum of matrix algebras equals the number of representations of the number $n$ in the form

where $n_{1} \geqslant n_{2} \geqslant \ldots \geqslant n_{k}>0$.