Olympiad problems

1999 Romania National Olympiad, 2

For a finite group $G$ we denote by $n(G)$ the number of elements of the group and by $s(G)$ the number of subgroups of it. Decide whether the following statements are true or false. a) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}<a.$ b) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}>a.$

1952 Miklós Schweitzer, 5

Tags: group theory , superior algebra

Let $ G$ be anon-commutative group. Consider all the one-to-one mappings $ a\rightarrow a'$ of $ G$ onto itself such that $ (ab)'\equal{}b'a'$ (i.e. the anti-automorphisms of $ G$). Prove that this mappings together with the automorphisms of $ G$ constitute a group which contains the group of the automorphisms of $ G$ as direct factor.

1993 Hungary-Israel Binational, 1

Tags: abstract algebra , group theory , superior algebra

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 $k \geq 2$ is an integer such that for all $x, y \in G$ and $i \in \{k-1, k, k+1\}$ the relation $(xy)^{i}= x^{i}y^{i}$ holds. Show that $G$ is Abelian.

2017 Romania National Olympiad, 3

Tags: abstract algebra , superior algebra , group theory , sylow subgroup

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]

2018 District Olympiad, 4

Tags: superior algebra , finite field

Let $n$ and $q$ be two natural numbers, $n\ge 2$, $q\ge 2$ and $q\not\equiv 1 (\text{mod}\ 4)$ and let $K$ be a finite field which has exactly $q$ elements. Show that for any element $a$ from $K$, there exist $x$ and $y$ in $K$ such that $a = x^{2^n} + y^{2^n}$. (Every finite field is commutative).

2018 Brazil Undergrad MO, 7

Tags: graph theory , isomorphism , superior algebra

Unless of isomorphisms, how many simple four-vertex graphs are there?

2011 IMC, 3

Tags: algebra , polynomial , superior algebra

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?

2012 Today's Calculation Of Integral, 826

Tags: abstract algebra , greatest common divisor , group theory , relatively prime , superior algebra , number theory

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 $\mathrm{coker} f$.

2003 District Olympiad, 3

Tags: algebra , polynomial , quadratic , superior algebra

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.

1993 Hungary-Israel Binational, 2

Tags: group theory , abstract algebra , superior algebra

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

2001 Romania National Olympiad, 2

Tags: superior algebra

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 $a\in A$.

2005 Alexandru Myller, 2

Tags: linear algebra , matrix , algebra , polynomial , superior algebra

Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$. [i]Dan Popescu[/i]

2007 Romania National Olympiad, 4

Tags: group theory , abstract algebra , superior algebra

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 $k$, $|S(k)|\leq n-1$.

1993 Hungary-Israel Binational, 4

Tags: group theory , abstract algebra , superior algebra

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 $H \leq G$ and $a, b \in G.$ Prove that $|aH \cap Hb|$ is either zero or a divisor of $|H |.$

1978 Miklós Schweitzer, 2

Tags: abstract algebra , superior algebra

For a distributive lattice $ L$, consider the following two statements: (A) Every ideal of $ L$ is the kernel of at least two different homomorphisms. (B) $ L$ contains no maximal ideal. Which one of these statements implies the other? (Every homomorphism $ \varphi$ of $ L$ induces an equivalence relation on $ L$: $ a \sim b$ if and only if $ a \varphi\equal{}b \varphi$. We do not consider two homomorphisms different if they imply the same equivalence relation.) [i]J. Varlet, E. Fried[/i]

2013 Miklós Schweitzer, 3

Tags: group theory , abstract algebra , superior algebra

Find for which positive integers $n$ the $A_n$ alternating group has a permutation which is contained in exactly one $2$-Sylow subgroup of $A_n$. [i]Proposed by Péter Pál Pálfy[/i]

2010 District Olympiad, 1

Tags: superior algebra

Let $ S$ be the sum of the inversible elements of a finite ring. Prove that $ S^2\equal{}S$ or $ S^2\equal{}0$.

2005 District Olympiad, 3

Tags: superior algebra

Let $(G,\cdot)$ be a group and let $F$ be the set of elements in the group $G$ of finite order. Prove that if $F$ is finite, then there exists a positive integer $n$ such that for all $x\in G$ and for all $y\in F$, we have \[ x^n y = yx^n. \]

1993 Hungary-Israel Binational, 6

Tags: group theory , abstract algebra , superior algebra

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 $a = b = 1.$

2014 Miklós Schweitzer, 6

Tags: vector , abstract algebra , superior algebra

Let $\rho:G\to GL(V)$ be a representation of a finite $p$-group $G$ over a field of characteristic $p$. Prove that if the restriction of the linear map $\sum_{g\in G} \rho(g)$ to a finite dimensional subspace $W$ of $V$ is injective, then the subspace spanned by the subspaces $\rho(g)W$ $(g\in G)$ is the direct sum of these subspaces.

2017 District Olympiad, 2

Tags: group theory , abstract algebra , superior algebra , morphism

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.

2000 IMC, 5

Tags: ring theory , superior algebra

Let $R$ be a ring of characteristic zero. Let $e,f,g\in R$ be idempotent elements (an element $x$ is called idempotent if $x^2=x$) satisfying $e+f+g=0$. Show that $e=f=g=0$.

1980 Miklós Schweitzer, 3

Tags: superior algebra

In a lattice, connected the elements $ a \wedge b$ and $ a \vee b$ by an edge whenever $ a$ and $ b$ are incomparable. Prove that in the obtained graph every connected component is a sublattice. [i]M. Ajtai[/i]

1997 Romania National Olympiad, 1

Tags: gaussian integer , superior algebra , rings , units

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 $A= \mathbb{Z}[i].$

1977 Miklós Schweitzer, 4

Tags: superior algebra

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]