Found problems: 250
2006 Iran MO (3rd Round), 2
$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$
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]
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]
2003 Gheorghe Vranceanu, 1
Prove that any permutation group of an order equal to a power of $ 2 $ contains a commutative subgroup whose order is the square of the exponent of the order of the group.
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 3
Let $A$ be Abelian group of order $p^4$, where $p$ is a prime number, and which has a subgroup $N$ with order $p$ such that $A/N\approx\mathbb{Z}/p^3\mathbb{Z}$. Find all $A$ expect isomorphic.
2011 Miklós Schweitzer, 4
Let G, H be two finite groups, and let $\varphi, \psi: G \to H$ be two surjective but non-injective homomorphisms. Prove that there exists an element of G that is not the identity element of G but whose images under $\varphi$ and $\psi$ are the same.
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$.
Gheorghe Țițeica 2025, P1
Let $G$ be a finite group and $a\in G$ a fixed element. Define the set $$S_a=\{g\in G\mid ga\neq ag, \,ga^2=a^2g\}.$$ Show that:
[list=a]
[*] if $g\in S_a$, then $ag^{-1}\in S_a$;
[*] $|S_a|$ is divisible by $4$.
1978 Germany Team Selection Test, 4
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
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 $\frac{n(G)}{s(G)}>a.$
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)
1954 Miklós Schweitzer, 7
[b]7.[/b] Find the finite groups having only one proper maximal subgroup. [b](A.12)[/b]
2007 IberoAmerican, 5
Let's say a positive integer $ n$ is [i]atresvido[/i] if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.
2005 Romania National Olympiad, 2
Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$, for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$).
a) Prove that $H^x=H^y$ if and only if $x^{-1}y \in H$;
b) Find the number of elements of the set $\bigcup_{x\in G} H^x$ as a function of $m$ and $n$.
[i]Calin Popescu[/i]
1996 Miklós Schweitzer, 4
Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$.
2012 Grigore Moisil Intercounty, 4
[b]a)[/b] Prove that for any two square matrices $ A,B $ of same order the equality $ \text{ord} (AB)=\text{ord} (BA) $ is true.
[b]b)[/b] Show that $ \text{ord} (ab) =\text{ord} (ba) $ if $ a,b $ are elements of a monoid and one of them is an unit.
PEN R Problems, 3
Prove no three lattice points in the plane form an equilateral triangle.
1994 USAMO, 2
The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, $\,\ldots, \,$ red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides
are red, blue, red, blue, red, blue, $\, \ldots, \,$ red, yellow, blue?
1987 Traian Lălescu, 2.3
Prove that $ C_G\left( N_G(H) \right)\subset N_G\left( C_G(H) \right) , $ for any subgroup $ H $ of $ G, $ and characterize the groups $ G $ for which equality in this relation holds for all $ H\le G. $
[i]Here,[/i] $ C_G,N_G $ [i]are the centralizer, respectively, the normalizer of[/i] $ G. $
1993 Hungary-Israel Binational, 5
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 $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?
2012 Centers of Excellency of Suceava, 1
Let be a natural number $ n\ge 2, $ a group $ G $ and two elements of it $ e_1,e_2 $ such that $ e_2e_1x=xe_2e_1, $ for any element $ x $ of $ G. $
Prove that $ \left( e_1xe_2 \right)^n =e_1x^ne_2, $ for any element $ x $ of $ G, $ if and only if $ e_2e_1=\left( e_2e_1\right)^n. $
[i]Ion Bursuc[/i]
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 $x,y$.
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 \}$
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.
2009 IberoAmerican Olympiad For University Students, 7
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).