Found problems: 103
2006 Romania National Olympiad, 3
Let $\displaystyle G$ be a finite group of $\displaystyle n$ elements $\displaystyle ( n \geq 2 )$ and $\displaystyle p$ be the smallest prime factor of $\displaystyle n$. If $\displaystyle G$ has only a subgroup $\displaystyle H$ with $\displaystyle p$ elements, then prove that $\displaystyle H$ is in the center of $\displaystyle G$.
[i]Note.[/i] The center of $\displaystyle G$ is the set $\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}$.
1979 Miklós Schweitzer, 4
For what values of $ n$ does the group $ \textsl{SO}(n)$ of all orthogonal transformations of determinant $ 1$ of the $ n$-dimensional Euclidean space possess a closed regular subgroup?($ \textsl{G}<\textsl{SO}(n)$ is called $ \textit{regular}$ if for any elements $ x,y$ of the unit sphere there exists a unique $ \varphi \in \textsl{G}$ such that $ \varphi(x)\equal{}y$.)
[i]Z. Szabo[/i]
1969 Miklós Schweitzer, 1
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]
1952 Miklós Schweitzer, 4
Let $ K$ be a finite field of $ p$ elements, where $ p$ is a prime. For every polynomial
$ f(x)\equal{}\sum_{i\equal{}0}^na_ix^i$ ($ \in K[x]$)
put
$ \overline{f(x)}\equal{}\sum_{i\equal{}0}^n a_ix^{p^i}$.
Prove that for any pair of polynomials $ f(x),g(x)\in K[x]$, $ \overline{f(x)}|\overline{g(x)}$ if and only if $ f(x)|g(x)$.
2014 IMS, 5
Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$, the number of group homomorphisms from $G_1$ to $H$ is equal to the number of group homomorphisms from $G_2$ to $H$. Prove that $G_1$ and $G_2$ are Isomorphic.
2009 IMS, 1
$ G$ is a group. Prove that the following are equivalent:
1. All subgroups of $ G$ are normal.
2. For all $ a,b\in G$ there is an integer $ m$ such that $ (ab)^m\equal{}ba$.
2008 District Olympiad, 3
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$.
2005 Romania National Olympiad, 4
Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let
\[ M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . \]
Prove that the following statements are equivalent:
a) $A$ is a field;
b) $M$ is not empty and the smallest element in $M$ is $2^n+1$.
[i]Marian Andronache[/i]
2013 District Olympiad, 2
Problem 2. A group $\left( G,\cdot \right)$ has the propriety$\left( P \right)$, if, for any
automorphism f for G,there are two automorphisms
g and h in G, so that $f\left( x \right)=g\left( x \right)\cdot h\left( x \right)$, whatever $x\in G$would be. Prove that:
(a) Every group which the property $\left( P \right)$ is comutative.
(b) Every commutative finite group of odd order doesn’t have the $\left( P \right)$ property.
(c) No finite group of order $4n+2,n\in \mathbb{N}$, doesn’t have the $\left( P \right)$property.
(The order of a finite group is the number of elements of that group).
2011 IMC, 3
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?
1993 Hungary-Israel Binational, 3
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.
1970 IMO Longlists, 28
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.
1985 Iran MO (2nd round), 6
In The ring $\mathbf R$, we have $\forall x \in \mathbf R : x^2=x$. Prove that in this ring
[b]i)[/b] Every element is equals to its additive inverse.
[b]ii)[/b] This ring has commutative property.
1967 Miklós Schweitzer, 3
Prove that if an infinite, noncommutative group $ G$ contains a proper normal subgroup with a commutative factor group, then $ G$ also contains an infinite proper normal subgroup.
[i]B. Csakany[/i]
2009 Miklós Schweitzer, 6
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$.
2010 Romania National Olympiad, 3
Let $G$ be a finite group of order $n$. Define the set
\[H=\{x:x\in G\text{ and }x^2=e\},\]
where $e$ is the neutral element of $G$. Let $p=|H|$ be the cardinality of $H$. Prove that
a) $|H\cap xH|\ge 2p-n$, for any $x\in G$, where $xH=\{xh:h\in H\}$.
b) If $p>\frac{3n}{4}$, then $G$ is commutative.
c) If $\frac{n}{2}<p\le\frac{3n}{4}$, then $G$ is non-commutative.
[i]Marian Andronache[/i]
1983 Miklós Schweitzer, 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]
2002 Romania National Olympiad, 4
Let $K$ be a field having $q=p^n$ elements, where $p$ is a prime and $n\ge 2$ is an arbitrary integer number. For any $a\in K$, one defines the polynomial $f_a=X^q-X+a$. Show that:
$a)$ $f=(X^q-X)^q-(X^q-X)$ is divisible by $f_1$;
$b)$ $f_a$ has at least $p^{n-1}$ essentially different irreducible factors $K[X]$.
2014 IMS, 7
Let $G$ be a finite group such that for every two subgroups of it like $H$ and $K$, $H \cong K$ or $H \subseteq K$ or $K \subseteq H$. Prove that we can produce each subgroup of $G$ with 2 elements at most.
2010 District Olympiad, 2
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.
2009 IMS, 7
Let $ G$ be a group such that $ G'$ is abelian and each normal and abelian subgroup of $ G$ is finite. Prove that $ G$ is finite.
1965 Miklós Schweitzer, 2
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$).
1973 Miklós Schweitzer, 2
Let $ R$ be an Artinian ring with unity. Suppose that every idempotent element of $ R$ commutes with every element of $ R$ whose square is $ 0$. Suppose $ R$ is the sum of the ideals $ A$ and $ B$. Prove that $ AB\equal{}BA$.
[i]A. Kertesz[/i]
2006 Pre-Preparation Course Examination, 3
a) If $K$ is a finite extension of the field $F$ and $K=F(\alpha,\beta)$ show that $[K: F]\leq [F(\alpha): F][F(\beta): F]$
b) If $gcd([F(\alpha): F],[F(\beta): F])=1$ then does the above inequality always become equality?
c) By giving an example show that if $gcd([F(\alpha): F],[F(\beta): F])\neq 1$ then equality might happen.
2009 Miklós Schweitzer, 4
Prove that the polynomial
\[ f(x) \equal{} \frac {x^n \plus{} x^m \minus{} 2}{x^{\gcd(m,n)} \minus{} 1}\]
is irreducible over $ \mathbb{Q}$ for all integers $ n > m > 0$.