Olympiad problems

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]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, A1

Tags: superior algebra , algebra , polynomial , galois theory , complex number

Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$

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]

2014 IMS, 3

Tags: superior algebra , vector , abstract algebra , ring theory

Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.

1970 Miklós Schweitzer, 1

Tags: superior algebra

We have $ 2n\plus{}1$ elements in the commutative ring $ R$: \[ \alpha,\alpha_1,...,\alpha_n,\varrho_1,...,\varrho_n .\] Let us define the elements \[ \sigma_k\equal{}k\alpha \plus{} \sum_{i\equal{}1}^n \alpha_i\varrho_i^k .\] Prove that the ideal $ (\sigma_0,\sigma_1,...,\sigma_k,...)$ can be finitely generated. [i]L. Redei[/i]

2015 District Olympiad, 4

Tags: ring theory , abstract algebra , nilpotence , superior algebra

Let $ m $ be a non-negative ineger, $ n\ge 2 $ be a natural number, $ A $ be a ring which has exactly $ n $ elements, and an element $ a $ of $ A $ such that $ 1-a^k $ is invertible, for all $ k\in\{ m+1,m+2,...,m+n-1\} . $ Prove that $ a $ is nilpotent.

2004 Gheorghe Vranceanu, 1

Tags: superior algebra , group theory , abstract algebra

Let $(G,\cdot)$ be a group, and let $H_1,H_2$ be proper subgroups s.t. $H_1\cap H_2=\{e\}$, where $e$ is the identity element of $G$. They also have the following properties: [b]i)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_1\setminus\{e\}\Rightarrow xy\in H_2$ [b]ii)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_2\setminus\{e\}\Rightarrow xy\in H_1$ Prove that: [b]a)[/b] $|H_1|=|H_2|$ [b]b)[/b] $|G|=|H_1|\cdot |H_2|$

2005 Alexandru Myller, 2

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

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]

1999 Romania National Olympiad, 4

Tags: superior algebra , integral domain , polynomial ring

Let $A$ be an integral domain and $A[X]$ be its associated ring of polynomials. For every integer $n \ge 2$ we define the map $\varphi_n : A[X] \to A[X],$ $\varphi_n(f)=f^n$ and we assume that the set $$M= \Big\{ n \in \mathbb{Z}_{\ge 2} : \varphi_n \mathrm{~is~an~endomorphism~of~the~ring~} A[X] \Big\}$$ is nonempty. Prove that there exists a unique prime number $p$ such that $M=\{p,p^2,p^3, \ldots\}.$

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 \[ x^n y = yx^n. \]

1985 Iran MO (2nd round), 4

Tags: group theory , abstract algebra , superior algebra

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 $G.$

2008 District Olympiad, 3

Tags: superior algebra , group theory , abstract algebra

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$.

1982 Miklós Schweitzer, 2

Tags: superior algebra

Consider the lattice of all algebraically closed subfields of the complex field $ \mathbb{C}$ whose transcendency degree (over $ \mathbb{Q}$) is finite. Prove that this lattice is not modular. [i]L. Babai[/i]

2014 IMS, 7

Tags: group theory , abstract algebra , superior algebra

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.

1989 Greece National Olympiad, 4

Tags: group theory , algebra , superior algebra

In a group $G$, we have two elements $x,y$ such that $x^{n}=e,y^2=e,yxy=x^{-1}$, $n\ge 1$. Prove that for any $k\in\mathbb{N}$ holds $(x^ky)^2=e$. Note : e=group's identity .

2014 Contests, 3

Tags: superior algebra , vector , abstract algebra , ring theory

Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.

1998 Romania National Olympiad, 2

Tags: group theory , abstract algebra , superior algebra

$\textbf{a) }$ Let $p \geq 2$ be a natural number and $G_p = \bigcup\limits_{n \in \mathbb{N}} \lbrace z \in \mathbb{C} \mid z^{p^n}=1 \rbrace.$ Prove that $(G_p, \cdot)$ is a subgroup of $(\mathbb{C}^*, \cdot).$ $\textbf{b) }$ Let $(H, \cdot)$ be an infinite subgroup of $(\mathbb{C}^*, \cdot).$ Prove that all proper subgroups of $H$ are finite if and only if $H=G_p$ for some prime $p.$

2014 Miklós Schweitzer, 6

Tags: superior algebra , vector , abstract 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.

1952 Miklós Schweitzer, 5

Tags: superior algebra , group theory

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.

2009 District Olympiad, 4

Tags: polynomial , superior algebra , finite field , probability

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

2009 IberoAmerican Olympiad For University Students, 7

Tags: group theory , abstract algebra , superior algebra

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).

2006 Romania National Olympiad, 3

Tags: superior algebra , group theory , abstract algebra

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 $\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}$.

1952 Miklós Schweitzer, 4

Tags: superior algebra , algebra , polynomial

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)$.

2010 Iran MO (3rd Round), 5

Tags: superior algebra , abstract algebra , linear algebra , matrix , group theory

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.

1993 Hungary-Israel Binational, 7

Tags: superior algebra , group theory , abstract 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. Assume $|G'| = 2$. Prove that $|G : G'|$ is even.