The group is abelian

Let $\mathcal{G}$ be a finite group such that $\left ( \left | \mathcal{G} \right |  , 3 \right ) =1$. If for the elements $a, \beta \in \mathcal{G}$ holds that

$$\left ( a \beta \right )^3 = a^3 \beta^3$$

then prove that $\mathcal{G}$ is abelian.

Solution

Least number $n$ so that embeds

Let ${\rm Gl}_2 \left(\mathbb{F_5} \right)$ be the group of invertible $2 \times 2$ matrices over $\mathbb{F}_5$ and $\mathcal{S}_n$ be the group of permutations of $n$ objects. What is the least $n \in \mathbb{N}$ such that there is an embedding of ${\rm Gl}_2 \left(\mathbb{F_5} \right)$ into $\mathcal{S}_n$ ?

Solution

Even permutation

Let $\alpha$ and $\beta$ be elements of $\mathcal{S}_n$. Prove that ${\alpha}^{-1}{\beta}^{-1}\alpha \beta$ is an even permutation.

Solution

Finite matrix group

Let $\mathcal{G}$ be a finite subgroup of ${\rm GL}_n (\mathbb{C})$ (this is the group of the $n \times n$ invertible matrices over $\mathbb{C}$). If $\sum \limits_{g \in \mathcal{G}} {\rm Tr}(g)=0$ then prove that $\sum \limits_{g \in \mathcal{G}} g =0$.

Solution

On some homomorphsims

Find all homomorphisms $\varphi$ in the following cases:

1. $\varphi:\mathbb{Q} \rightarrow \mathbb{Q}$
2. $\varphi:\mathbb{Q}[\sqrt{2}] \rightarrow \mathbb{Q}[\sqrt{3}]$
3. $\varphi:\mathbb{Z}[i]\rightarrow \mathbb{C}$

Solution 

Non commutative ring but commutative quotient

Give an example of a non commutative ring $\mathcal{R}$ which has an ideal $I$ and $\mathcal{R}/I$ is commutative.

Solution

An ideal of $\mathbb{Z}[i]$

Give an example of an ideal of $\mathbb{Z}[i]$ known as the ring of Gaussian integers.

Solution

Endomorphism and abelian group

Let $(\mathcal{G}, \cdot)$ be a group for which there exist an endomorphism $f:\mathcal{G} \rightarrow \mathcal{G}$ such that

$$f\left ( x^n y^{n+1} \right )= x^{n+1}y^n \quad \text{forall} \; x, y \in \mathcal{G}$$

Prove that $\mathcal{G}$ is abelian.

Solution

Von Neumann ring

Let $(\mathcal{R}, +, \cdot)$ be a ring without necessary a unitary element but it has at least two elements. If $\mathcal{R}$ furthermore satisfies the property: "for every element $a \in \mathcal{R}$ that is not zero ($a \neq 0$) there exists a unique element $b \in \mathcal{R}$ such that $aba=a$ , then prove that:

1. $\mathcal{R}$ does not have zero divisors.
2. $bab=b$.
3. $\mathcal{R}$ has a unitary element.
4. $\mathcal{R}$ is a divisor ring.

Solution

An exercise on a ring

Let $\mathcal{R}$ be a ring and we suppose that $a, b \in \mathcal{R}$ be two elements of $\mathcal{R}$ such that $ab=1$. Prove that:

1. $a^n b^n =1$ forall $n \geq 1$.
2. $(1-ba)b^n =0$ forall $n \geq 1$.
3. the elements $ba$ and $1-ba$ are idempotent and orthogonal.
4. if $c \in \mathcal{R}$ and $ca=1$ , then the element $a$ is invertible and it holds that:
   
   $$a^{-1}=b=c$$

Solution

Commutative ring

Let $\mathcal{R}$ such that $x^3=x$ forall $x \in \mathcal{R}$. Prove that $\mathcal{R}$ is commutative.

Solution

Cyclic group

Let $\mathcal{G}$ be a group of order $n$. Prove that if $\gcd(\varphi(n), n )=1$ then $\mathcal{G}$ is cyclic, where $\varphi$ denotes Euler's totient function.

Solution

Normal subgroup

Let $\mathcal{S}_4$ denote the symmetric $4$ - permutation group. Prove that

$$\mathcal{H}=\big\{{{\rm{id}},\,({1\,2})\,({3\,4}),\,({1\,3})\,({2\,4}),\,({1\,4})\,({2\,3})}\big\}$$

is a normal subgroup of $\mathcal{S}_4$.

Solution

They are isomorphic

Prove that:

$$\mathbb{Z}_2 \times \mathbb{Z}_4 \times \mathbb{Z}_8 \bigg/\langle [1]_4, [2]_4, [4]_8 \rangle \cong \mathbb{Z}_4 \times \mathbb{Z}_8$$

Solution