Found problems: 250
Gheorghe Țițeica 2024, P2
a) Let $n$ be a positive integer $G$ be a a group with $|G|<\frac{4n^2}{n-\varphi(n)}$. Suppose that $Z(G)$ contains at least $\varphi(n)+1$ elements of order $n$. Prove that $G$ is abelian.
b) Find a noncommutative group $G$ with $16$ elements such that $Z(G)$ contains two elements of order two.
[i]Robert Rogozsan & Filip Munteanu[/i]
2021 Alibaba Global Math Competition, 16
Let $G$ be a finite group, and let $H_1, H_2 \subset G$ be two subgroups. Suppose that for any representation of $G$ on a finite-dimensional complex vector space $V$, one has that
\[\text{dim} V^{H_1}=\text{dim} V^{H_2},\]
where $V^{H_i}$ is the subspace of $H_i$-invariant vectors in $V$ ($i=1,2$). Prove that
\[Z(G) \cap H_1=Z(G) \cap H_2.\]
Here $Z(G)$ denotes the center of $G$.
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]
2010 Iran MO (3rd Round), 3
suppose that $G<S_n$ is a subgroup of permutations of $\{1,...,n\}$ with this property that for every $e\neq g\in G$ there exist exactly one $k\in \{1,...,n\}$ such that $g.k=k$. prove that there exist one $k\in \{1,...,n\}$ such that for every $g\in G$ we have $g.k=k$.(20 points)
1977 Spain Mathematical Olympiad, 2
Prove that all square matrices of the form (with $a, b \in R$),
$$\begin{pmatrix}
a & b \\
-b & a
\end{pmatrix}$$
form a commutative field $K$ when considering the operations of addition and matrix product. Prove also that if $A \in K$ is an element of said field, there exist two matrices of $K$ such that the square of each is equal to $A$.
2010 Iran MO (3rd Round), 2
prove the third sylow theorem: suppose that $G$ is a group and $|G|=p^em$ which $p$ is a prime number and $(p,m)=1$. suppose that $a$ is the number of $p$-sylow subgroups of $G$ ($H<G$ that $|H|=p^e$). prove that $a|m$ and $p|a-1$.(Hint: you can use this: every two $p$-sylow subgroups are conjugate.)(20 points)
2011 Bogdan Stan, 2
Solve the system
$$ \left\{\begin{matrix} ax=b\\bx=a \end{matrix}\right. $$
independently of the fixed elements $ a,b $ of a group of odd order.
[i]Marian Andronache[/i]
2006 Germany Team Selection Test, 3
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
2023 Olympic Revenge, 6
We say that $H$ permeates $G$ if $G$ and $H$ are finite groups and for all subgroup $F$ of $G$ there is $H'\cong H$ with $H'\le F$ or $F\le H'\le G$. Suppose that a non-abelian group $H$ permeates $G$ and let $S=\langle H'\le G | H'\cong H\rangle$. Show that
$$|\bigcap_{H'\in S} H'|>1$$
2014 European Mathematical Cup, 2
In each vertex of a regular $n$-gon $A_1A_2...A_n$ there is a unique pawn. In each step it is allowed:
1. to move all pawns one step in the clockwise direction or
2. to swap the pawns at vertices $A_1$ and $A_2$.
Prove that by a fi nite series of such steps it is possible to swap the pawns at vertices:
a) $A_i$ and $A_{i+1}$ for any $ 1 \leq i < n$ while leaving all other pawns in their initial place
b) $A_i$ and $A_j$ for any $ 1 \leq i < j \leq n$ leaving all other pawns in their initial place.
[i]Proposed by Matija Bucic[/i]
2006 Cezar Ivănescu, 2
Prove that the set $ \left\{ \left. \begin{pmatrix} \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2}\end{pmatrix}\right| x\in\mathbb{R}^{*} \right\} $ along with the usual multiplication of matrices form a group, determine an isomorphism between this group and the group of multiplicative real numbers.
2013 Princeton University Math Competition, 2
How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips.
1957 Miklós Schweitzer, 10
[b]10.[/b] An Abelian group $G$ is said to have the property $(A)$ if torsion subgroup of $G$ is a direct summand of $G$. Show that if $G$ is an Abelian group such that $nG$ has the property $(A)$ for some positive integer $n$, then $G$ itself has the property $(A)$. [b](A. 13)[/b]
2007 Gheorghe Vranceanu, 1
Let $ M $ denote the set of the primitives of a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $
[b]ii)[/b] Show that $ M $ along with the operation $ *:M^2\longrightarrow M $ defined as $ F*G=F+G(2007) $ form a commutative group.
[b]iii)[/b] Show that $ M $ is isomorphic with the additive group of real numbers.
1974 Spain Mathematical Olympiad, 5
Let $(G, \cdot )$ be a group and $e$ an identity element. Prove that if all elements $x$ of $G$ satisfy $x\cdot x = e$ then $(G, \cdot)$ is abelian (that is, commutative).
1958 Miklós Schweitzer, 1
[b]1.[/b] Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) [b](A. 14)[/b]
2004 Gheorghe Vranceanu, 1
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] $|G|=|H_1|\cdot |H_2|$
2012 IMC, 3
Given an integer $n>1$, let $S_n$ be the group of permutations of the numbers $1,\;2,\;3,\;\ldots,\;n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_n$. It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group $S_n$. The player who made the last move loses the game. The first move is made by A. Which player has a winning strategy?
[i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]
2020 CHMMC Winter (2020-21), 1
[i](5 pts)[/i] Let $n$ be a positive integer, $K = \{1, 2, \dots, n\}$, and $\sigma : K \rightarrow K$ be a function with the property that $\sigma(i) = \sigma(j)$ if and only if $i = j$ (in other words, $\sigma$ is a \textit{bijection}). Show that there is a positive integer $m$ such that
\[ \underbrace{\sigma(\sigma( \dots \sigma(i) \dots ))}_\textrm{$m$ times} = i \]
for all $i \in K$.
2021 Alibaba Global Math Competition, 13
Let $M_n=\{(u,v) \in S^n \times S^n: u \cdot v=0\}$, where $n \ge 2$, and $u \cdot v$ is the Euclidean inner product of $u$ and $v$. Suppose that the topology of $M_n$ is induces from $S^n \times S^n$.
(1) Prove that $M_n$ is a connected regular submanifold of $S^n \times S^n$.
(2) $M_n$ is Lie Group if and only if $n=2$.
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 $G.$
2018 Brazil Undergrad MO, 25
Consider the $ \mathbb {Z} / (10) $ additive group automorphism group of integers module $10$, that is,
$ A = \left \{\phi: \mathbb {Z} / (10) \to \mathbb {Z} / (10) | \phi-automorphism \right \}$
2023 Miklós Schweitzer, 5
Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity.
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$.
2007 Pre-Preparation Course Examination, 2
Let $\{A_{1},\dots,A_{k}\}$ be matrices which make a group under matrix multiplication. Suppose $M=A_{1}+\dots+A_{k}$. Prove that each eigenvalue of $M$ is equal to $0$ or $k$.