Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.

For a group $\left( G, \star \right)$ and $A, B$ two non-void subsets of $G$, we define $A \star B = \left\{ a \star b : a \in A \ \text{and}\ b \in B \right\}$. (a) Prove that if $n \in \mathbb N, \, n \geq 3$, then the group $\left( \mathbb Z \slash n \mathbb Z,+\right)$ can be writen as $\mathbb Z \slash n \mathbb Z = A+B$, where $A, B$ are two non-void subsets of $\mathbb Z \slash n \mathbb Z$ and $A \neq \mathbb Z \slash n \mathbb Z, \, B \neq \mathbb Z \slash n \mathbb Z, \, \left| A \cap B \right| = 1$. (b) If $\left( G, \star \right)$ is a finite group, $A, B$ are two subsets of $G$ and $a \in G \setminus \left( A \star B \right)$, then prove that function $f : A \to G \setminus B$ given by $f(x) = x^{-1}\star a$ is well-defined and injective. Deduce that if $|A|+|B| > |G|$, then $G = A \star B$. [hide="Question."]Does the last result have a name?[/hide]

Let $ \Psi\equal{}\langle A;...\rangle$ be an arbitrary, countable algebraic structure (that is, $ \Psi$ can have an arbitrary number of finitary operations and relations). Prove that $ \Psi$ has as many as continuum automorphisms if and only if for any finite subset $ A'$ of $ A$ there is an automorphism $ \pi_{A'}$ of $ \Psi$ different from the identity automorphism and such that \[ (x) \pi_{A'}\equal{}x\] for every $ x \in A'$. [i]M. Makkai[/i]

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 $a_1, a_2,...,a_{51}$ be non-zero elements of a field of characteristic $p$. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_1, ... , b_{51}$. If this new sequence is a permutation of the original one, find all possible values of $p$.

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. Show that every element of $S_{n}$ is a product of $2$-cycles.

A set $G$ with elements $u,v,w...$ is a Group if the following conditions are fulfilled: $(\text{i})$ There is a binary operation $\circ$ defined on $G$ such that $\forall \{u,v\}\in G$ there is a $w\in G$ with $u\circ v = w$. $(\text{ii})$ This operation is associative; i.e. $(u\circ v)\circ w = u\circ (v\circ w)$ $\forall\{u,v,w\}\in G$. $(\text{iii})$ $\forall \{u,v\}\in G$, there exists an element $x\in G$ such that $u\circ x = v$, and an element $y\in G$ such that $y\circ u = v$. Let $K$ be a set of all real numbers greater than $1$. On $K$ is defined an operation by $ a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}$. Prove that $K$ is a Group.

Let $A$ be a ring. $a)$ Show that the set $Z(A)=\{a\in A|ax=xa,\ \text{for all}\ x\in A\}$ is a subring of the ring $A$. $b)$ Prove that, if any commutative subring of $A$ is a field, then $A$ is a field.

Let $\left( A,+, \cdot \right)$ be a ring that verifies the following properties: (i) it has a unit, $1$, and its order is $p$, a prime number; (ii) there is $B \subset A, \, |B| = p$, such that: for all $x,y \in A$, there is $b \in B$ such that $xy = byx$. Prove that $A$ is commutative. [i]Ion Savu[/i]

Let $ S$ be a semigroup without proper two-sided ideals and suppose that for every $ a,b \in S$ at least one of the products $ ab$ and $ ba$ is equal to one of the elements $ a,b$. Prove that either $ ab\equal{}a$ for all $ a,b \in S$ or $ ab\equal{}b$ for all $ a,b \in S$. [i]L. Megyesi[/i]

Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$, for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$). a) Prove that $H^x=H^y$ if and only if $x^{-1}y \in H$; b) Find the number of elements of the set $\bigcup_{x\in G} H^x$ as a function of $m$ and $n$. [i]Calin Popescu[/i]

Determine all non-trivial finite rings with am unit element in which the sum of all elements is invertible. [i]Mihai Opincariu[/i]

Let $p$ be a prime number, $n$ a natural number which is not divisible by $p$, and $\mathbb{K}$ is a finite field, with $char(K) = p, |K| = p^n, 1_{\mathbb{K}}$ unity element and $\widehat{0} = 0_{\mathbb{K}}.$ For every $m \in \mathbb{N}^{*}$ we note $ \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} $ and define the polynomial \[ f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X]. \] a) Show that roots of $f_1$ are $ \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}$. b) Let $m \in \mathbb{N}^{*}.$ Determine the set of roots from $\mathbb{K}$ of polynomial $f_{m}.$

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. Let $a, b \in G.$ Suppose that $ab^{2}= b^{3}a$ and $ba^{2}= a^{3}b.$ Prove that $a = b = 1.$

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]

Let $ \mathcal G$ be the set of all finite groups with at least two elements. a) Prove that if $ G\in \mathcal G$, then the number of morphisms $ f: G\to G$ is at most $ \sqrt [p]{n^n}$, where $ p$ is the largest prime divisor of $ n$, and $ n$ is the number of elements in $ G$. b) Find all the groups in $ \mathcal G$ for which the inequality at point a) is an equality.

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]

A set system $ (S,L)$ is called a Steiner triple system, if $ L\neq\emptyset$, any pair $ x,y\in S$, $ x\neq y$ of points lie on a unique line $ \ell\in L$, and every line $ \ell\in L$ contains exactly three points. Let $ (S,L)$ be a Steiner triple system, and let us denote by $ xy$ the thrid point on a line determined by the points $ x\neq y$. Let $ A$ be a group whose factor by its center $ C(A)$ is of prime power order. Let $ f,h: S\to A$ be maps, such that $ C(A)$ contains the range of $ f$, and the range of $ h$ generates $ A$. Show, that if \[ f(x) \equal{} h(x)h(y)h(x)h(xy)\] holds for all pairs $ x\neq y$ of points, then $ A$ is commutative, and there exists an element $ k\in A$, such that $ f(x) \equal{} kh(x)$ for all $ x\in S$.

Let $ G$ be a group such that if $ a,b\in \mathbb{G}$ and $ a^2b\equal{}ba^2$, then $ ab\equal{}ba$. i)If $ G$ has $ 2^n$ elements, prove that $ G$ is abelian. ii) Give an example of a non-abelian group with $ G$'s property from the enounce.

Let be a ring that has the property that all its elements are the product of two idempotent elements of it. Show that: [b]a)[/b] $ 1 $ is the only unit of this ring. [b]b)[/b] this ring is Boolean.

Let $G$ a $n$ elements group. Find all the functions $f:G\rightarrow \mathbb{N}^*$ such that: (a) $f(x)=1$ if and only if $x$ is $G$'s identity; (b) $f(x^k)=\frac{f(x)}{(f(x),k)}$ for any divisor $k$ of $n$, where $(r,s)$ stands for the greatest common divisor of the positive integers $r$ and $s$.

Let $A$ be a commutative ring with $0 \neq 1$ such that for any $x \in A \setminus \{0\}$ there exist positive integers $m,n$ such that $(x^m+1)^n=x.$ Prove that any endomorphism of $A$ is an automorphism.

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 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 $a \in G, a^{n-1}\in Z (G).$

Let $ I$ be an ideal of the ring $ R$ and $ f$ a nonidentity permutation of the set $ \{ 1,2,\ldots, k \}$ for some $ k$. Suppose that for every $ 0 \not\equal{} a \in R, \;aI \not\equal{} 0$ and $ Ia \not\equal{}0$ hold; furthermore, for any elements $ x_1,x_2,\ldots ,x_k \in I$, \[ x_1x_2\ldots x_k\equal{}x_{1f}x_{2f}\ldots x_{kf}\] holds. Prove that $ R$ is commutative. [i]R. Wiegandt[/i]