What is the value of the series $\sum_{1 \leq l <m<n} \frac{1}{5^l3^m2^n}$

Let $ R$ be a finite commutative ring. Prove that $ R$ has a multiplicative identity element $ (1)$ if and only if the annihilator of $ R$ is $ 0$ (that is, $ aR\equal{}0, \;a\in R $ imply $ a\equal{}0$).

Let $K$ be a finite field and $f:K\to K^*$. Prove that there is a reducible polynomial $P\in K[X]$ s.t. $P(x)=f(x),\forall x\in K$. [i]Marian Andronache[/i]

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 $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?

Prove that if $ a,x,y$ are $ p$-adic integers different from $ 0$ and $ p | x, pa | xy$, then \[ \frac 1y \frac{(1\plus{}x)^y\minus{}1}{x} \equiv \frac{\log (1\plus{}x)}{x} \;\;\;\; ( \textrm{mod} \; a\ ) \\\\ .\] [i]L. Redei[/i]

Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$. (a) Prove that $f_n$ is well defined. (b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective. [i]Bogdan Enescu[/i]

Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.

Consider the endomorphism ring of an Abelian torsion-free (resp. torsion) group $ G$. Prove that this ring is Neumann-regular if and only if $ G$ is a discrete direct sum of groups isomorphic to the additive group of the rationals (resp. ,a discrete direct sum of cyclic groups of prime order). (A ring $ R$ is called Neumann-regular if for every $ \alpha \in R$ there exists a $ \beta \in R$ such that $ \alpha \beta \alpha\equal{}\alpha$.) [i]E. Freid[/i]

Let $G$ be a group such that every subgroup of $G$ is subnormal. Suppose that there exists $N$ normal subgroup of $G$ such that $Z(N)$ is nontrivial and $G/N$ is cyclic. Prove that $Z(G)$ is nontrivial. ($Z(G)$ denotes the center of $G$). [b]Note[/b]: A subgroup $H$ of $G$ is subnormal if there exist subgroups $H_1,H_2,\ldots,H_m=G$ of $G$ such that $H\lhd H_1\lhd H_2 \lhd \ldots \lhd H_m= G$ ($\lhd$ denotes normal subgroup).

Let $G$ be a hyper elementary abelian $p-$group and let $f : G \rightarrow G$ be a homomorphism. Then prove that $\ker f$ is isomorphic to $\mathrm{coker} f$.

Let be a group and two coprime natural numbers $ m,n. $ Show that if the applications $ G\ni x\mapsto x^{m+1},x^{n+1} $ are surjective endomorphisms, then the group is commutative.

suppose that $p$ is a prime number. find that smallest $n$ such that there exists a non-abelian group $G$ with $|G|=p^n$. SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries. the exam time was 5 hours.

Let \[ f(x)\equal{}a_0\plus{}a_1x\plus{}a_2x^2\plus{}a_{10}x^{10}\plus{}a_{11}x^{11}\plus{}a_{12}x^{12}\plus{}a_{13}x^{13} \; (a_{13} \not\equal{}0) \] and \[ g(x)\equal{}b_0\plus{}b_1x\plus{}b_2x^2\plus{}b_{3}x^{3}\plus{}b_{11}x^{11}\plus{}b_{12}x^{12}\plus{}b_{13}x^{13} \; (b_{3} \not\equal{}0) \] be polynomials over the same field. Prove that the degree of their greatest common divisor is at least $ 6$. [i]L. Redei[/i]

Let $\mathcal K$ be a field of characteristic $p$, $p \equiv 1 \left( \bmod 4 \right)$. (a) Prove that $-1$ is the square of an element from $\mathcal K.$ (b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$, of elements from $\mathcal K$. (c) Can $0$ be written in the same way? [i]Marian Andronache[/i]

Let $ G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $ G$ is finite. [i]J. Pelikan[/i]

Let $(G,\cdot)$ be a group with no elements of order 4, and let $f:G\rightarrow G$ be a group morphism such that $f(x)\in\{x,x^{-1}\}$, for all $x\in G$. Prove that either $f(x)=x$ for all $x\in G$, or $f(x)=x^{-1}$ for all $x\in G$.

Let $G$ be a group and $n \ge 2$ be an integer. Let $H_1, H_2$ be $2$ subgroups of $G$ that satisfy $$[G: H_1] = [G: H_2] = n \text{ and } [G: (H_1 \cap H_2)] = n(n-1).$$ Prove that $H_1, H_2$ are conjugate in $G.$ Official definitions: $[G:H]$ denotes the index of the subgroup of $H,$ i.e. the number of distinct left cosets $xH$ of $H$ in $G.$ The subgroups $H_1, H_2$ are conjugate if there exists $g \in G$ such that $g^{-1} H_1 g = H_2.$

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]

Let $\alpha \in \mathbb{C} \setminus \mathbb{Q}$ be such that the set $A= \{ a+b \alpha : a,b \in \mathbb{Z} \}$ is a ring with respect to the usual operations of $\mathbb{C}.$ If the ring $A$ has exactly four invertible elements, prove that $A= \mathbb{Z}[i].$

Let $n\geq 3$ be an integer and $S_{n}$ the permutation group. $G$ is a subgroup of $S_{n}$, generated by $n-2$ transpositions. For all $k\in\{1,2,\ldots,n\}$, denote by $S(k)$ the set $\{\sigma(k) \ : \ \sigma\in G\}$. Show that for any $k$, $|S(k)|\leq n-1$.

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]

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.

Let $G$ be a finite group and let $x_1,…,x_n$ be an enumeration of its elements. We consider the matrix $(a_{ij})_{1 \le i,j \le n},$ where $a_{ij}=0$ if $x_ix_j^{-1}=x_jx_i^{-1},$ and $a_{ij}=1$ otherwise. Find the parity of the integer $\det(a_{ij}).$

Show that the set of all elements minus $ 0 $ of a finite division ring that has at least $ 4 $ elements can be partitioned into two nonempty sets $ A,B $ having the property that $$ \sum_{x\in A} x=\prod_{y\in B} y. $$

Let $ G$ be an infinite compact topological group with a Hausdorff topology. Prove that $ G$ contains an element $ g \not\equal{} 1$ such that the set of all powers of $ g$ is either everywhere dense in $ G$ or nowhere dense in $ G$. [i]J. Erdos[/i]