Found problems: 250
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]
2006 VJIMC, Problem 2
Let $(G,\cdot)$ be a finite group of order $n$. Show that each element of $G$ is a square if and only if $n$ is odd.
2016 Postal Coaching, 5
Is it possible to define an operation $\star$ on $\mathbb Z$ such that[list=a][*] for any $a, b, c$ in $\mathbb Z, (a \star b) \star c = a \star (b \star c)$ holds;
[*] for any $x, y$ in $\mathbb Z, x \star x \star y = y \star x \star x=y$?[/list]
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.
2017 Brazil National Olympiad, 6.
[b]6.[/b] Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer.
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.
2007 IberoAmerican Olympiad For University Students, 6
Let $F$ be a field whose characteristic is not $2$, let $F^*=F\setminus\left\{0\right\}$ be its multiplicative group and let $T$ be the subgroup of $F^*$ constituted by its finite order elements. Prove that if $T$ is finite, then $T$ is cyclic and its order is even.
1994 Hungary-Israel Binational, 4
An [i]$ n\minus{}m$ society[/i] is a group of $ n$ girls and $ m$ boys. Prove that there exists numbers $ n_0$ and $ m_0$ such that every [i]$ n_0\minus{}m_0$ society[/i] contains a subgroup of five boys and five girls with the following property: either all of the boys know all of the girls or none of the boys knows none of the girls.
1989 Putnam, B2
Let S be a non-empty set with an associative operation that is left and right cancellative (xy=xz implies y=z, and yx = zx implies y = z). Assume that for every a in S the set {a^n : n = 0,1,2...} is finite. Must S be a group?
I haven't had much group theory at this point...
2011 VJIMC, Problem 4
Let $a,b,c$ be elements of finite order in some group. Prove that if $a^{-1}ba=b^2$, $b^{-2}cb^2=c^2$, and $c^{-3}ac^3=a^2$ then $a=b=c=e$, where $e$ is the unit element.
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 $i \in K$.
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]
1952 Miklós Schweitzer, 5
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.
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 $G$.
2008 Romania National Olympiad, 4
Let $ \mathcal G$ be the set of all finite groups with at least two elements.
a) Prove that if $ G\in \mathcal G$, then the number of morphisms $ f: G\to G$ is at most $ \sqrt [p]{n^n}$, where $ p$ is the largest prime divisor of $ n$, and $ n$ is the number of elements in $ G$.
b) Find all the groups in $ \mathcal G$ for which the inequality at point a) is an equality.
2008 Putnam, B4
Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$
2004 Silk Road, 4
Natural $n \geq 2$ is given. Group of people calls $n-compact$, if for any men from group, we can found $n$ people (without he), each two of there are familiar.
Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.
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]
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
$$|\bigcap_{H'\in S} H'|>1$$
1969 Putnam, B2
Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if "two' is replaced by "three'?
2014 USA Team Selection Test, 3
For a prime $p$, a subset $S$ of residues modulo $p$ is called a [i]sum-free multiplicative subgroup[/i] of $\mathbb F_p$ if
$\bullet$ there is a nonzero residue $\alpha$ modulo $p$ such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$ (all considered mod $p$), and
$\bullet$ there are no $a,b,c \in S$ (not necessarily distinct) such that $a+b \equiv c \pmod p$.
Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb F_p$ such that $\left\lvert S \right\rvert \ge N$.
[i]Proposed by Noga Alon and Jean Bourgain[/i]
2014 Romania National Olympiad, 4
Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $
[b]a)[/b] Prove that the order of $ G $ is a power of $ p. $
[b]b)[/b] Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $
2011 District Olympiad, 2
Let $ G $ be the set of matrices of the form $ \begin{pmatrix} a&b\\0&1 \end{pmatrix} , $ with $ a,b\in\mathbb{Z}_7,a\neq 0. $
[b]a)[/b] Verify that $ G $ is a group.
[b]b)[/b] Show that $ \text{Hom}\left( (G,\cdot) ; \left( \mathbb{Z}_7,+ \right) \right) =\{ 0\} $
2007 IMS, 1
Suppose there exists a group with exactly $n$ subgroups of index 2. Prove that there exists a finite abelian group $G$ that has exactly $n$ subgroups of index 2.
1999 Miklós Schweitzer, 4
A permutation f of the set of integers is called bounded if | x - f (x) | is bounded. Bounded permutations with permutation multiplication form a group W. Show that the additive group of rational numbers is not isomorphic to any subgroup of W.