Found problems: 103
1972 Miklós Schweitzer, 4
Let $ G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $ G$ is finite.
[i]J. Pelikan[/i]
1998 IMC, 2
Consider the following statement: for any permutation $\pi_1\not=\mathbb{I}$ of $\{1,2,...,n\}$ there is a permutation $\pi_2$ such that any permutation on these numbers can be obtained by a finite compostion of $\pi_1$ and $\pi_2$.
(a) Prove the statement for $n=3$ and $n=5$.
(b) Disprove the statement for $n=4$.
2002 Romania National Olympiad, 1
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.
2012 District Olympiad, 3
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 $s$.
1977 Miklós Schweitzer, 4
Let $ p>5$ be a prime number. Prove that every algebraic integer of the $ p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field.
[i]K. Gyory[/i]
2009 IMS, 2
Let $ R$ be a ring with 1. Every element in $ R$ can be written as product of idempotent ($ u^n\equal{}u$ for some $ n$) elements. Prove that $ R$ is commutative
2003 District Olympiad, 3
Let $\displaystyle \mathcal K$ be a finite field such that the polynomial $\displaystyle X^2-5$ is irreducible over $\displaystyle \mathcal K$. Prove that:
(a) $1+1 \neq 0$;
(b) for all $\displaystyle a \in \mathcal K$, the polynomial $\displaystyle X^5+a$ is reducible over $\displaystyle \mathcal K$.
[i]Marian Andronache[/i]
[Edit $1^\circ$] I wanted to post it in "Superior Algebra - Groups, Fields, Rings, Ideals", but I accidentally put it here :blush: Can any mod move it? I'd be very grateful.
[Edit $2^\circ$] OK, thanks.
2012 Today's Calculation Of Integral, 826
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 $\mathrm{coker} f$.
2012 Romania National Olympiad, 2
[color=darkred] Let $(R,+,\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\in R$ , where $[a,b]=ab-ba$ , $a,b\in R$ . Prove that:
[list]
[b]a)[/b] $[x,f(y)]=[f(x),y]$ and $x[x,y]=f(x)[x,y]$ , for any $x,y\in R\ ;$
[b]b)[/b] If $R$ is a division ring and $f$ is different from the identity function, then $R$ is commutative.
[/list]
[/color]
1974 Miklós Schweitzer, 4
Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic.
[i]L. Lovasz, J. Pelikan[/i]
2007 District Olympiad, 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]
2001 Romania National Olympiad, 2
Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\ge 1$, such that $a^m=a^p$ for all $a\in A$.
1969 Miklós Schweitzer, 2
Let $ p\geq 7$ be a prime number, $ \zeta$ a primitive $ p$th root of unity, $ c$ a rational number. Prove that in the additive group generated by the numbers $ 1,\zeta,\zeta^2,\zeta^3\plus{}\zeta^{\minus{}3}$ there are only finitely many elements whose norm is equal to $ c$. (The norm is in the $ p$th cyclotomic field.)
[i]K. Gyory[/i]
2014 District Olympiad, 3
Let $(A,+,\cdot)$ be an unit ring with the property: for all $x\in A$,
\[ x+x^{2}+x^{3}=x^{4}+x^{5}+x^{6} \]
[list=a]
[*]Let $x\in A$ and let $n\geq2$ be an integer such that $x^{n}=0$. Prove that $x=0$.
[*]Prove that $x^{4}=x$, for all $x\in A$.[/list]
1962 Miklós Schweitzer, 2
Determine the roots of unity in the field of $ p$-adic numbers.
[i]L. Fuchs[/i]
1967 Miklós Schweitzer, 1
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]
2007 Romania National Olympiad, 3
Let $n\geq 1$ be an integer. Find all rings $(A,+,\cdot)$ such that all $x\in A\setminus\{0\}$ satisfy $x^{2^{n}+1}=1$.
1968 Miklós Schweitzer, 6
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]
2013 Miklós Schweitzer, 4
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]
1985 Iran MO (2nd round), 4
Let $G$ be a group and let $a$ be a constant member of it. Prove that
\[G_a = \{x | \exists n \in \mathbb Z , x=a^n\}\]
Is a subgroup of $G.$
1963 Miklós Schweitzer, 3
Let $ R\equal{}R_1\oplus R_2$ be the direct sum of the rings $ R_1$ and $ R_2$, and let $ N_2$ be the annihilator ideal of $ R_2$ (in $ R_2$). Prove that $ R_1$ will be an ideal in every ring $ \widetilde{R}$ containing $ R$ as an ideal if and only if the only homomorphism from $ R_1$ to $ N_2$ is the zero homomorphism. [Gy. Hajos]
1977 Miklós Schweitzer, 3
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]
2001 District Olympiad, 1
For any $n\in \mathbb{N}^*$, let $H_n=\left\{\frac{k}{n!}\ |\ k\in \mathbb{Z}\right\}$.
a) Prove that $H_n$ is a subgroup of the group $(Q,+)$ and that $Q=\bigcup_{n\in \mathbb{N}^*} H_n$;
b) Prove that if $G_1,G_2,\ldots, G_m$ are subgroups of the group $(Q,+)$ and $G_i\neq Q,\ (\forall) 1\le i\le m$, then
$G_1\cup G_2\cup \ldots \cup G_m\neq Q$
[i]Marian Andronache & Ion Savu[/i]
1977 Miklós Schweitzer, 5
Suppose that the automorphism group of the finite undirected graph $ X\equal{}(P, E)$ is isomorphic to the quaternion group (of order $ 8$). Prove that the adjacency matrix of $ X$ has an eigenvalue of multiplicity at least $ 4$.
($ P\equal{} \{ 1,2,\ldots, n \}$ is the set of vertices of the graph $ X$. The set of edges $ E$ is a subset of the set of all unordered pairs of elements of $ P$. The group of automorphisms of $ X$ consists of those permutations of $ P$ that map edges to edges. The adjacency matrix $ M\equal{}[m_{ij}]$ is the $ n \times n$ matrix defined by $ m_{ij}\equal{}1$ if $ \{ i,j \} \in E$ and $ m_{i,j}\equal{}0$ otherwise.)
[i]L. Babai[/i]
1981 Miklós Schweitzer, 4
Let $ G$ be finite group and $ \mathcal{K}$ a conjugacy class of $ G$ that generates $ G$. Prove that the following two statements are equivalent:
(1) There exists a positive integer $ m$ such that every element of $ G$ can be written as a product of $ m$ (not necessarily distinct) elements of $ \mathcal{K}$.
(2) $ G$ is equal to its own commutator subgroup.
[i]J. Denes[/i]