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]

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

Let $\mathcal K$ be a field of characteristic $p$, $p \equiv 1 \left( \bmod 4 \right)$. (a) Prove that $-1$ is the square of an element from $\mathcal K.$ (b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$, of elements from $\mathcal K$. (c) Can $0$ be written in the same way? [i]Marian Andronache[/i]

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.

Determine the roots of unity in the field of $ p$-adic numbers. [i]L. Fuchs[/i]

Let $ A$ and $ B$ be two Abelian groups, and define the sum of two homomorphisms $ \eta$ and $ \chi$ from $ A$ to $ B$ by \[ a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ .\] With this addition, the set of homomorphisms from $ A$ to $ B$ forms an Abelian group $ H$. Suppose now that $ A$ is a $ p$-group ( $ p$ a prime number). Prove that in this case $ H$ becomes a topological group under the topology defined by taking the subgroups $ p^kH \;(k\equal{}1,2,...)$ as a neighborhood base of $ 0$. Prove that $ H$ is complete in this topology and that every connected component of $ H$ consists of a single element. When is $ H$ compact in this topology? [L. Fuchs]

A ring $ A $ has property [i](P),[/i] if $ A $ is finite and there exists $ (\{ 0\}\neq R,+)\le (A,+) $ such that $ (U(A),\cdot )\cong (R,+) . $ Show that: [b]a)[/b] If a ring has property [i](P),[/i] then, the number of its elements is even. [b]b)[/b] There are infinitely many rings of distinct order that have property [i](P).[/i]

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.

a) If $K$ is a finite extension of the field $F$ and $K=F(\alpha,\beta)$ show that $[K: F]\leq [F(\alpha): F][F(\beta): F]$ b) If $gcd([F(\alpha): F],[F(\beta): F])=1$ then does the above inequality always become equality? c) By giving an example show that if $gcd([F(\alpha): F],[F(\beta): F])\neq 1$ then equality might happen.

Prove that if $A$ is a commutative finite ring with at least two elements and $n$ is a positive integer, then there exists a polynomial of degree $n$ with coefficients in $A$ which does not have any roots in $A$.

Let $ G$ and $ H$ be countable Abelian $ p$-groups ($ p$ an arbitrary prime). Suppose that for every positive integer $ n$, \[ p^nG \not\equal{} p^{n\plus{}1}G .\] Prove that $ H$ is a homomorphic image of $ G$. [i]M. Makkai[/i]

In The ring $\mathbf R$, we have $\forall x \in \mathbf R : x^2=x$. Prove that in this ring [b]i)[/b] Every element is equals to its additive inverse. [b]ii)[/b] This ring has commutative property.

Let $r,s,t$ positive integers which are relatively prime and $a,b \in G$, $G$ a commutative multiplicative group with unit element $e$, and $a^r=b^s=(ab)^t=e$. (a) Prove that $a=b=e$. (b) Does the same hold for a non-commutative group $G$?

Let be a group $ G $ of order $ 1+p, $ where $ p $ is and odd prime. Show that if $ p $ divides the number of automorphisms of $ G, $ then $ p\equiv 3\pmod 4. $

If A,B are invertible and the set {A^k - B^k | k is a natural number} is finite , then there exists a natural number m such that A^m = B^m.

[b]a)[/b] Show that, for any distinct natural numbers $ m,n, $ the rings $ \mathbb{Z}_2\times \underbrace{\cdots}_{m\text{ times}} \times\mathbb{Z}_2,\mathbb{Z}_2\times \underbrace{\cdots}_{n\text{ times}} \times\mathbb{Z}_2 $ are homomorphic, but not isomorphic. [b]b)[/b] Show that there are infinitely many pairwise nonhomomorphic rings of same order.

A set $G$ with elements $u,v,w...$ is a Group if the following conditions are fulfilled: $(\text{i})$ There is a binary operation $\circ$ defined on $G$ such that $\forall \{u,v\}\in G$ there is a $w\in G$ with $u\circ v = w$. $(\text{ii})$ This operation is associative; i.e. $(u\circ v)\circ w = u\circ (v\circ w)$ $\forall\{u,v,w\}\in G$. $(\text{iii})$ $\forall \{u,v\}\in G$, there exists an element $x\in G$ such that $u\circ x = v$, and an element $y\in G$ such that $y\circ u = v$. Let $K$ be a set of all real numbers greater than $1$. On $K$ is defined an operation by $ a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}$. Prove that $K$ is a Group.

Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.

Show that there is an unique group $ G $ (up to isomorphism) of order $ 1986 $ which has the property that there is at most one subgroup of it having order $ n, $ for every natural number $ n. $

Let $ R$ be an Artinian ring with unity. Suppose that every idempotent element of $ R$ commutes with every element of $ R$ whose square is $ 0$. Suppose $ R$ is the sum of the ideals $ A$ and $ B$. Prove that $ AB\equal{}BA$. [i]A. Kertesz[/i]

Consider the following statement: for any permutation $\pi_1\not=\mathbb{I}$ of $\{1,2,...,n\}$ there is a permutation $\pi_2$ such that any permutation on these numbers can be obtained by a finite compostion of $\pi_1$ and $\pi_2$. (a) Prove the statement for $n=3$ and $n=5$. (b) Disprove the statement for $n=4$.

Let $(G,\cdot)$ be a group with no elements of order 4, and let $f:G\rightarrow G$ be a group morphism such that $f(x)\in\{x,x^{-1}\}$, for all $x\in G$. Prove that either $f(x)=x$ for all $x\in G$, or $f(x)=x^{-1}$ for all $x\in G$.

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. Show that every element of $S_{n}$ is a product of $2$-cycles.

Let $n$ be a positive integer and $G$ be a finite group of order $n.$ A function $f:G \to G$ has the $(P)$ property if $f(xyz)=f(x)f(y)f(z)~\forall~x,y,z \in G.$ $\textbf{(a)}$ If $n$ is odd, prove that every function having the $(P)$ property is an endomorphism. $\textbf{(b)}$ If $n$ is even, is the conclusion from $\textbf{(a)}$ still true?

Let be two nontrivial rings linked by an application ($ K\stackrel{\vartheta }{\mapsto } L $) having the following properties: $ \text{(i)}\quad x,y\in K\implies \vartheta (x+y) = \vartheta (x) +\vartheta (y) $ $ \text{(ii)}\quad \vartheta (1)=1 $ $ \text{(iii)}\quad \vartheta \left( x^3\right) =\vartheta^3 (x) $ [b]a)[/b] Show that if $ \text{char} (L)\ge 4, $ and $ K,L $ are fields, then $ \vartheta $ is an homomorphism. [b]b)[/b] Prove that if $ K $ is a noncommutative division ring, then it’s possible that $ \vartheta $ is not an homomorphism.