Found problems: 250
1996 Miklós Schweitzer, 4
Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$.
1993 Hungary-Israel Binational, 1
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.
2019 LIMIT Category C, Problem 5
Let $G=(S^1,\cdot)$ be a group. Then its nontrivial subgroups
$\textbf{(A)}~\text{are necessarily finite}$
$\textbf{(B)}~\text{can be infinite}$
$\textbf{(C)}~\text{can be dense in }S^1$
$\textbf{(D)}~\text{None of the above}$
2022 District Olympiad, P2
Let $(G,\cdot)$ be a group and $H\neq G$ be a subgroup so that $x^2=y^2$ for all $x,y\in G\setminus H.$ Show that $(H,\cdot)$ is an Abelian group.
2002 District Olympiad, 1
Let $ A $ be a ring, $ a\in A, $ and let $ n,k\ge 2 $ be two natural numbers such that $ n\vdots\text{char} (A) $ and $ 1+a=a^k. $ Show that the following propositions are true:
[b]a)[/b] $ \forall s\in\mathbb{N}\quad \exists p_0,p_1,\ldots ,p_{k-1}\in\mathbb{Z}_{\ge 0}\quad a^s=\sum_{i=0}^{k-1} p_ia^{i} . $
[b]b)[/b] $ \text{ord} (a)\neq\infty . $
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.
2010 AMC 12/AHSME, 25
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $ 32$?
$ \textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255$
1995 Brazil National Olympiad, 2
Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$, and $f(xy) = f(x) f(y)$ for all $x,y$.
1980 Miklós Schweitzer, 5
Let $ G$ be a transitive subgroup of the symmetric group $ S_{25}$ different from $ S_{25}$ and $ A_{25}$. Prove that the order of $ G$ is not divisible by $ 23$.
[i]J. Pelikan[/i]
2017 Romania National Olympiad, 3
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]
2005 IMC, 6
6) $G$ group, $G_{m}$ and $G_{n}$ commutative subgroups being the $m$ and $n$ th powers of the elements in $G$. Prove $G_{gcd(m,n)}$ is commutative.
PEN H Problems, 49
Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.
1996 Romania National Olympiad, 1
Prove that a group $G$ in which exactly two elements other than the identity commute with each other is isomorphic to $\mathbb{Z}/3 \mathbb{Z}$ or $S_3.$
2019 District Olympiad, 3
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 $\det(a_{ij}).$
2008 IMS, 5
Prove that there does not exist a ring with exactly 5 regular elements.
($ a$ is called a regular element if $ ax \equal{} 0$ or $ xa \equal{} 0$ implies $ x \equal{} 0$.)
A ring is not necessarily commutative, does not necessarily contain unity element, or is not necessarily finite.
2009 Indonesia TST, 3
Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.
2006 District Olympiad, 2
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.
2023 Romania National Olympiad, 1
Let $(G, \cdot)$ a finite group with order $n \in \mathbb{N}^{*},$ where $n \geq 2.$ We will say that group $(G, \cdot)$ is arrangeable if there is an ordering of its elements, such that
\[
G = \{ a_1, a_2, \ldots, a_k, \ldots , a_n \} = \{ a_1 \cdot a_2, a_2 \cdot a_3, \ldots, a_k \cdot a_{k + 1}, \ldots , a_{n} \cdot a_1 \}.
\]
a) Determine all positive integers $n$ for which the group $(Z_n, +)$ is arrangeable.
b) Give an example of a group of even order that is arrangeable.
1985 Miklós Schweitzer, 6
Determine all finite groups $G$ that have an automorphism $f$ such that $H\not\subseteq f(H)$ for all proper subgroups $H$ of $G$. [B. Kovacs]
2004 Nicolae Coculescu, 3
Let be a finite group $ G $ having an endomorphism $ \eta $ that has exactly one fixed point.
[b]a)[/b] Demonstrate that the function $ f:G\longrightarrow G $ defined as $ f(x)=x^{-1}\cdot\eta (x) $ is bijective.
[b]b)[/b] Show that $ G $ is commutative if the composition of the function $ f $ from [b]a)[/b] with itself is the identity function.
2016 USA Team Selection Test, 1
Let $S = \{1, \dots, n\}$. Given a bijection $f : S \to S$ an [i]orbit[/i] of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$. We denote by $c(f)$ the number of distinct orbits of $f$. For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$.
Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that \[ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) \] where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$.
[i]Proposed by Maria Monks Gillespie[/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]
2003 Romania National Olympiad, 4
[b]a)[/b] Prove that the sum of all the elements of a finite union of sets of elements of finite cyclic subgroups of the group of complex numbers, is an integer number.
[b]b)[/b] Show that there are finite union of sets of elements of finite cyclic subgroups of the group of complex numbers such that the sum of all its elements is equal to any given integer.
[i]Paltin Ionescu[/i]
2021 Science ON grade XII, 4
Consider a group $G$ with at least $2$ elements and the property that each nontrivial element has infinite order. Let $H$ be a cyclic subgroup of $G$ such that the set $\{xH\mid x\in G\}$ has $2$ elements. \\
$\textbf{(a)}$ Prove that $G$ is cyclic. \\
$\textbf{(b)}$ Does the conclusion from $\textbf{(a)}$ stand true if $G$ contains nontrivial elements of finite order?
2011 Gheorghe Vranceanu, 1
[b]a)[/b] Let $ B,A $ be two subsets of a finite group $ G $ such that $ |A|+|B|>|G| . $ Show that $ G=AB. $
[b]b)[/b] Show that the cyclic group of order $ n+1 $ is the product of the sets $ \{ 0,1,2,\ldots ,m \} $ and $ \{ m,m+1,m+2,\ldots ,n\} , $ where $ 0,1,2,\ldots n $ are residues modulo $ n+1 $ and $ m\le n. $