Found problems: 250
2005 Gheorghe Vranceanu, 1
For a natural number $ n\ge 2, $ prove that the $ \text{n-ary} $ direct product of the group of order $ 2 $ is abelian and isomorphic with the group of the power set of a set under symmetric difference.
2001 District Olympiad, 1
For any $n\in \mathbb{N}^*$, let $H_n=\left\{\frac{k}{n!}\ |\ k\in \mathbb{Z}\right\}$.
a) Prove that $H_n$ is a subgroup of the group $(Q,+)$ and that $Q=\bigcup_{n\in \mathbb{N}^*} H_n$;
b) Prove that if $G_1,G_2,\ldots, G_m$ are subgroups of the group $(Q,+)$ and $G_i\neq Q,\ (\forall) 1\le i\le m$, then
$G_1\cup G_2\cup \ldots \cup G_m\neq Q$
[i]Marian Andronache & Ion Savu[/i]
1985 Traian Lălescu, 1.3
Let $ G $ be a finite group of odd order having, at least, three elements. For $ a\in G $ denote $ n(a) $ as the number of ways $ a $ can be written as a product of two distinct elements of $ G. $
Prove that $ \sum_{\substack{a\in G\\a\neq\text{id}}} n(a) $ is a perfect square.
2007 Grigore Moisil Intercounty, 4
Consider the group $ \{f:\mathbb{C}\setminus\mathbb{Q}\longrightarrow\mathbb{C}\setminus\mathbb{Q} | f\text{ is bijective}\} $ under the composition of functions. Find the order of the smallest subgroup of it that:
$ \text{(1)} $ contains the function $ z\mapsto \frac{z-1}{z+1} . $
$ \text{(2)} $ contains the function $ z\mapsto \frac{z-3}{z+1} . $
$ \text{(3)} $ contain both of the above functions.
2002 Iran Team Selection Test, 2
$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.
2013 Miklós Schweitzer, 3
Find for which positive integers $n$ the $A_n$ alternating group has a permutation which is contained in exactly one $2$-Sylow subgroup of $A_n$.
[i]Proposed by Péter Pál Pálfy[/i]
1993 Hungary-Israel Binational, 7
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.
2023 District Olympiad, P2
Let $(G,\cdot)$ be a grup with neutral element $e{}$, and let $H{}$ and $K$ be proper subgroups of $G$, satisfying $H\cap K=\{e\}$. It is known that $(G\setminus(H\cup K))\cup\{e\}$ is closed under the operation of $G$. Prove that $x^2=e$ for all the elements $x{}$ of $G{}$.
2011 District Olympiad, 4
Let be a ring $ A. $ Denote with $ N(A) $ the subset of all nilpotent elements of $ A, $ with $ Z(A) $ the center of $ A, $ and with $ U(A) $ the units of $ A. $ Prove:
[b]a)[/b] $ Z(A)=A\implies N(A)+U(A)=U(A) . $
[b]b)[/b] $ \text{card} (A)\in\mathbb{N}\wedge a+U(A)\subset U(A)\implies a\in N(A) . $
2010 District Olympiad, 2
Let $ G$ be a group such that if $ a,b\in \mathbb{G}$ and $ a^2b\equal{}ba^2$, then $ ab\equal{}ba$.
i)If $ G$ has $ 2^n$ elements, prove that $ G$ is abelian.
ii) Give an example of a non-abelian group with $ G$'s property from the enounce.
2011 Indonesia TST, 2
At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).
2019 Romania National Olympiad, 2
Let $n \geq 4$ be an even natural number and $G$ be a subgroup of $GL_2(\mathbb{C})$ with $|G| = n.$ Prove that there exists $H \leq G$ such that $\{ I_2 \} \neq H$ and $H \neq G$ such that $XYX^{-1} \in H, \: \forall X \in G$ and $\forall Y \in H$
1998 Romania National Olympiad, 2
$\textbf{a) }$ 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 $p.$
1960 Putnam, A7
Let $N(n)$ denote the smallest positive integer $N$ such that $x^N =e$ for every element $x$ of the symmetric group $S_n$, where $e$ denotes the identity permutation. Prove that if $n>1,$
$$\frac{N(n)}{N(n-1)} =\begin{cases} p \;\text{if}\; n\; \text{is a power of a prime } p\\
1\; \text{otherwise}.
\end{cases}$$
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]
2013 Miklós Schweitzer, 4
Let $A$ be an Abelian group with $n$ elements. Prove that there are two subgroups in $\text{GL}(n,\Bbb{C})$, isomorphic to $S_n$, whose intersection is isomorphic to the automorphism group of $A$.
[i]Proposed by Zoltán Halasi[/i]
1968 Putnam, B2
Let $G$ be a finite group with $n$ elements and $K$ a subset of $G$ with more than $\frac{n}{2}$ elements. Show that for any $g\in G$ one can find $h,k\in K$ such that $g=h\cdot k$.
1997 Miklós Schweitzer, 7
Let G be an abelian group, $0\leq\varepsilon<1$ and $f : G\to\Bbb R^n$ a function that satisfies the inequality.
$$||f(x+y)-f(x)-f(y)|| \leq \varepsilon ||f (y)|| \qquad (x, y)\in G^2$$
Prove that there is an additive function $A : G\to \Bbb R^n$ and a continuous function $\varphi : A (G) \to\Bbb R^n$ such that $f = \varphi\circ A$.
2000 Romania National Olympiad, 3
We say that the abelian group $ G $ has property [i](P)[/i] if, for any commutative group $ H, $ any $ H’\le H $ and any momorphism $ \mu’:H\longrightarrow G, $ there exists a morphism $ \mu :H\longrightarrow G $ such that $ \mu\bigg|_{H’} =\mu’ . $ Show that:
[b]a)[/b] the group $ \left( \mathbb{Q}^*,\cdot \right) $ hasn’t property [i](P).[/i]
[b]b)[/b] the group $ \left( \mathbb{Q}, +\right) $ has property [i](P).[/i]
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 $k$.
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.
2011 Indonesia TST, 2
At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).
2006 Iran MO (3rd Round), 3
$L$ is a fullrank lattice in $\mathbb R^{2}$ and $K$ is a sub-lattice of $L$, that $\frac{A(K)}{A(L)}=m$. If $m$ is the least number that for each $x\in L$, $mx$ is in $K$. Prove that there exists a basis $\{x_{1},x_{2}\}$ for $L$ that $\{x_{1},mx_{2}\}$ is a basis for $K$.
2012 Putnam, 2
Let $*$ be a commutative and associative binary operation on a set $S.$ Assume that for every $x$ and $y$ in $S,$ there exists $z$ in $S$ such that $x*z=y.$ (This $z$ may depend on $x$ and $y.$) Show that if $a,b,c$ are in $S$ and $a*c=b*c,$ then $a=b.$
1989 IMO Longlists, 82
Let $ A$ be a set of positive integers such that no positive integer greater than 1 divides all the elements of $ A.$ Prove that any sufficiently large positive integer can be written as a sum of elements of $ A.$ (Elements may occur several times in the sum.)