Let $n \geq 4$ be an even natural number and $G$ be a subgroup of $GL_2(\mathbb{C})$ with $|G| = n.$ Prove that there exists $H \leq G$ such that $\{ I_2 \} \neq H$ and $H \neq G$ such that $XYX^{-1} \in H, \: \forall X \in G$ and $\forall Y \in H$

$\textbf{a) }$ Let $p \geq 2$ be a natural number and $G_p = \bigcup\limits_{n \in \mathbb{N}} \lbrace z \in \mathbb{C} \mid z^{p^n}=1 \rbrace.$ Prove that $(G_p, \cdot)$ is a subgroup of $(\mathbb{C}^*, \cdot).$ $\textbf{b) }$ Let $(H, \cdot)$ be an infinite subgroup of $(\mathbb{C}^*, \cdot).$ Prove that all proper subgroups of $H$ are finite if and only if $H=G_p$ for some prime $p.$

Prove that a nontrivial finite ring is not a skew field if and only if the equation $ x^n+y^n=z^n $ has nontrivial solutions in this ring for any natural number $ n. $

Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.

Let $ G$ be an infinite group generated by nilpotent normal subgroups. Prove that every maximal Abelian normal subgroup of $ G$ is infinite. (We call an Abelian normal subgroup maximal if it is not contained in another Abelian normal subgroup.) [i]P. Erdos[/i]

Let $A$ be an Abelian group with $n$ elements. Prove that there are two subgroups in $\text{GL}(n,\Bbb{C})$, isomorphic to $S_n$, whose intersection is isomorphic to the automorphism group of $A$. [i]Proposed by Zoltán Halasi[/i]

Let $(A,+,\cdot)$ a 9 elements ring. Prove that the following assertions are equivalent: (a) For any $x\in A\backslash\{0\}$ there are two numbers $a\in \{-1,0,1\}$ and $b\in \{-1,1\}$ such that $x^2+ax+b=0$. (b) $(A,+,\cdot)$ is a field.

Let $p$ be a prime number. Call a positive integer $n$ interesting if \[x^n-1=(x^p-x+1)f(x)+pg(x)\] for some polynomials $f$ and $g$ with integer coefficients. a) Prove that the number $p^p-1$ is interesting. b) For which $p$ is $p^p-1$ the minimal interesting number?

Let $R$ be a ring where $\forall a\in R: a^2=0$. Prove that $abc+abc=0$ for all $a,b,c\in R$.

We say that the abelian group $ G $ has property [i](P)[/i] if, for any commutative group $ H, $ any $ H’\le H $ and any momorphism $ \mu’:H\longrightarrow G, $ there exists a morphism $ \mu :H\longrightarrow G $ such that $ \mu\bigg|_{H’} =\mu’ . $ Show that: [b]a)[/b] the group $ \left( \mathbb{Q}^*,\cdot \right) $ hasn’t property [i](P).[/i] [b]b)[/b] the group $ \left( \mathbb{Q}, +\right) $ has property [i](P).[/i]

Let $G$ be the permutation group of a finite set $\Omega$.Consider $S\subset G$ such that $1\in S$ and for any $x,y\in \Omega$ there exists a unique element $\sigma \in S$ such that $\sigma (x)=y$.Prove that,if the elements of $S \setminus \{1\}$ are conjugate in $G$,then $G$ is $2-$transitive on $\Omega$

In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group. Suppose that $n \geq 1$ is such that the mapping $x \mapsto x^{n}$ from $G$ to itself is an isomorphism. Prove that for each $a \in G, a^{n-1}\in Z (G).$

For which commutative finite groups is the product of all elements equal to the unit element?

Let $ A$ and $ B$ be two Abelian groups, and define the sum of two homomorphisms $ \eta$ and $ \chi$ from $ A$ to $ B$ by \[ a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ .\] With this addition, the set of homomorphisms from $ A$ to $ B$ forms an Abelian group $ H$. Suppose now that $ A$ is a $ p$-group ( $ p$ a prime number). Prove that in this case $ H$ becomes a topological group under the topology defined by taking the subgroups $ p^kH \;(k\equal{}1,2,...)$ as a neighborhood base of $ 0$. Prove that $ H$ is complete in this topology and that every connected component of $ H$ consists of a single element. When is $ H$ compact in this topology? [L. Fuchs]

A group $ G $ of order at least $ 4 $ has the property that there exists a natural number $ n\not\in\{ 1,|G| \} $ such that $ G $ admits exactly $ \binom{|G|-1}{n-1} $ subgroups of order $ n. $ Show that $ G $ is commutative. [i]Marius Tărnăuceanu[/i]

Let $R$ be a noncommutative finite ring with multiplicative identity element $1$. Show that if the subring generated by $I \cup \{1\}$ is $R$ for each nonzero ideal $I$ then $R$ is simple.

Let $A=(a_{ij})_{i, j=1}^n$ be a symmetric $n\times n$ matrix with real entries, and let $\lambda _1, \lambda _2, \dots, \lambda _n$ denote its eigenvalues. Show that $$\sum_{1\le i<j\le n} a_{ii}a_{jj}\ge \sum_{1\le i < j\le n} \lambda _i \lambda _j$$ and determine all matrices for which equality holds. (Proposed by Matrin Niepel, Comenius University, Bratislava)

Let $ A$ be a commutative unitary ring with an odd number of elements. Prove that the number of solutions of the equation $ x^2 \equal{} x$ (in $ A$) divides the number of invertible elements of $ A$.

Let be a group $ G $ of order $ 1+p, $ where $ p $ is and odd prime. Show that if $ p $ divides the number of automorphisms of $ G, $ then $ p\equiv 3\pmod 4. $

In a certain ring there are as many units as there are nilpotent elements. Prove that the order of the ring is a power of $ 2. $ [i]Dinu Şerbănescu[/i]

Let $ K $ be the group of Klein. Prove that: [b]a)[/b] There is an unique division ring (up to isomorphism), $ D, $ such that $ (D,+)\cong K. $ [b]b)[/b] There are no division rings $ A $ such that $ (A\setminus\{ 0\} ,+)\cong K. $

Let $ A $ be a ring that is not a division ring, and such that any non-unit of it is idempotent. Show that: [b]a)[/b] $ \left( U(A) +A\setminus\left( U(A)\cup \{ 0\} \right) \right)\cap U(A) =\emptyset . $ [b]b)[/b] Every element of $ A $ is idempotent.

Let $ G$ be a transitive subgroup of the symmetric group $ S_{25}$ different from $ S_{25}$ and $ A_{25}$. Prove that the order of $ G$ is not divisible by $ 23$. [i]J. Pelikan[/i]

Let $G$ be a finite group with the following property: If $f$ is an automorphism of $G$, then there exists $m\in\mathbb{N^\star}$, so that $f(x)=x^{m} $ for all $x\in G$. Prove that G is commutative. [i]Marian Andronache[/i]

We say that a ring $A$ has property $(P)$ if any non-zero element can be written uniquely as the sum of an invertible element and a non-invertible element. a) If in $A$, $1+1=0$, prove that $A$ has property $(P)$ if and only if $A$ is a field. b) Give an example of a ring that is not a field, containing at least two elements, and having property $(P)$. [i]Dan Schwarz[/i]