Found problems: 183
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]
Let $K$ be a finite field and $f:K\to K^*$. Prove that there is a reducible polynomial $P\in K[X]$ s.t. $P(x)=f(x),\forall x\in K$.
[i]Marian Andronache[/i]
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.
Let be a finite field $ K. $ Say that two polynoms $ f,g $ from $ K[X] $ are [i]neighbours,[/i] if the have the same degree and they differ by exactly one coefficient.
[b]a)[/b] Show that all the neighbours of $ 1+X^2 $ from $ \mathbb{Z}_3[X] $ are reducible in $ \mathbb{Z}_3[X] . $
[b]b)[/b] If $ |K|\ge 4, $ show that any polynomial of degree $ |K|-1 $ from $ K[X] $ has a neighbour from $ K[X] $ that is reducible in $ K[X] , $ and also has a neighbour that doesn´t have any root in $ K. $
Let $R$ be a noncommutative finite ring with multiplicative identity element $1$. Show that if the subring generated by $I \cup \{1\}$ is $R$ for each nonzero ideal $I$ then $R$ is simple.
Determine all finite groups $G$ that have an automorphism $f$ such that $H\not\subseteq f(H)$ for all proper subgroups $H$ of $G$. [B. Kovacs]
Let $G$ be a group and $n \ge 2$ be an integer. Let $H_1, H_2$ be $2$ subgroups of $G$ that satisfy $$[G: H_1] = [G: H_2] = n \text{ and } [G: (H_1 \cap H_2)] = n(n-1).$$ Prove that $H_1, H_2$ are conjugate in $G.$
Official definitions: $[G:H]$ denotes the index of the subgroup of $H,$ i.e. the number of distinct left cosets $xH$ of $H$ in $G.$ The subgroups $H_1, H_2$ are conjugate if there exists $g \in G$ such that $g^{-1} H_1 g = H_2.$
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]
Prove that any identity that holds for every finite $ n$-distributive lattice also holds for the lattice of all convex subsets of the $ (n\minus{}1)$-dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice $ n$-$ \textit{distributive}$ if \[ x \wedge (\bigvee_{i\equal{}0}^n y_i)\equal{}\bigvee_{j\equal{}0}^n(x \wedge (\bigvee_{0\leq i \leq n, \;i \not\equal{} j\ }y_i))\] holds for all elements of the lattice.)
[i]A. Huhn[/i]
Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let
\[ M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . \]
Prove that the following statements are equivalent:
a) $A$ is a field;
b) $M$ is not empty and the smallest element in $M$ is $2^n+1$.
[i]Marian Andronache[/i]
Let $ K$ be a finite field of $ p$ elements, where $ p$ is a prime. For every polynomial
$ f(x)\equal{}\sum_{i\equal{}0}^na_ix^i$ ($ \in K[x]$)
put
$ \overline{f(x)}\equal{}\sum_{i\equal{}0}^n a_ix^{p^i}$.
Prove that for any pair of polynomials $ f(x),g(x)\in K[x]$, $ \overline{f(x)}|\overline{g(x)}$ if and only if $ f(x)|g(x)$.
[color=darkred] Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients.
[/color]
Let $\displaystyle \mathcal K$ be a finite field such that the polynomial $\displaystyle X^2-5$ is irreducible over $\displaystyle \mathcal K$. Prove that:
(a) $1+1 \neq 0$;
(b) for all $\displaystyle a \in \mathcal K$, the polynomial $\displaystyle X^5+a$ is reducible over $\displaystyle \mathcal K$.
[i]Marian Andronache[/i]
[Edit $1^\circ$] I wanted to post it in "Superior Algebra - Groups, Fields, Rings, Ideals", but I accidentally put it here :blush: Can any mod move it? I'd be very grateful.
[Edit $2^\circ$] OK, thanks.
Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\ge 1$, such that $a^m=a^p$ for all $a\in A$.
Let $\left( A,+, \cdot \right)$ be a ring that verifies the following properties:
(i) it has a unit, $1$, and its order is $p$, a prime number;
(ii) there is $B \subset A, \, |B| = p$, such that: for all $x,y \in A$, there is $b \in B$ such that $xy = byx$.
Prove that $A$ is commutative.
[i]Ion Savu[/i]
$\textbf{a) }$ 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 $p.$
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.
Determine the roots of unity in the field of $ p$-adic numbers.
[i]L. Fuchs[/i]
Problem 2. A group $\left( G,\cdot \right)$ has the propriety$\left( P \right)$, if, for any
automorphism f for G,there are two automorphisms
g and h in G, so that $f\left( x \right)=g\left( x \right)\cdot h\left( x \right)$, whatever $x\in G$would be. Prove that:
(a) Every group which the property $\left( P \right)$ is comutative.
(b) Every commutative finite group of odd order doesn’t have the $\left( P \right)$ property.
(c) No finite group of order $4n+2,n\in \mathbb{N}$, doesn’t have the $\left( P \right)$property.
(The order of a finite group is the number of elements of that group).
For any $k \in \mathbb{Z},$ define $$F_k=X^4+2(1-k)X^2+(1+k)^2.$$ Find all values $k \in \mathbb{Z}$ such that $F_k$ is irreducible over $\mathbb{Z}$ and reducible over $\mathbb{Z}_p,$ for any prime $p.$
[i]Marius Vladoiu[/i]
Let $\rho:G\to GL(V)$ be a representation of a finite $p$-group $G$ over a field of characteristic $p$. Prove that if the restriction of the linear map $\sum_{g\in G} \rho(g)$ to a finite dimensional subspace $W$ of $V$ is injective, then the subspace spanned by the subspaces $\rho(g)W$ $(g\in G)$ is the direct sum of these subspaces.
Prove that the intersection of all maximal left ideals of a ring is a (two-sided) ideal.
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.
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.
Suppose $k \geq 2$ is an integer such that for all $x, y \in G$ and $i \in \{k-1, k, k+1\}$ the relation $(xy)^{i}= x^{i}y^{i}$ holds. Show that $G$ is Abelian.
In a certain ring there are as many units as there are nilpotent elements. Prove that the order of the ring is a power of $ 2. $
[i]Dinu Şerbănescu[/i]