Action of a group on a set. Relation of conjugacy

$G$ is the group of all nonsingular linear operators;

$G$ is the group of orthogonal operators;

$G$ is the group of operators whose matrices in the basis $\left(e_{1}, \ldots , e_{n}\right)$ are diagonal;

$G$ is the group of operators whose matrices in the basis $\left(e_{1}, \ldots , e_{n}\right)$ are upper triangular.

$G$ is the group from 57.1, (c);

$G$ is the group from 57.1, (d).

$G$ is the group of all orthogonal operators in a three-dimensional Euclidean space;

$G$ is the group of all proper orthogonal operators in a two-dimensional Euclidean space.

Find the orbits of the group $G$ in $X$.

Let $e_{1}, \ldots , e_{n}$ be a basis of $V$ such that $e_{1}, \ldots , e_{k}$ is a basis of some subspace $U$. Find, in the basis $e_{1}, \ldots , e_{n}$, the matrices of the operators from the stabilizer $G_{U}$.

Find the orbits of $G$ in $F$.

Let $e_{i} \in V_{i} \backslash V_{i-1}, i=1, \ldots , n$. Prove that $e_{1}, \ldots , e_{n}$ is a basis of $V$.

Find, in the basis $e_{1}, \ldots , e_{n}$, the matrices of the operators from the stabilizer $G_{f}$.

Find the orbits of the action of $G$ in $X$ and $Y$.

Find the stabilizer $G_{a}$ of a decomposable $q$-vector $a$ (a vector in $S^{q}(V)$).

Prove that this defines an action of $G$ on $B$.

Describe the orbits of $G$ in $B$. Find their number.

Describe the stabilizer $G_{b}$ of a positive definite function $b$.

Prove that this defines an action of $G$ on $L(V)$.

Describe the orbits of $G$ in $L(V)$.

$g=\left(\begin{array}{cccccccccc}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 5 & 8 & 3 & 9 & 4 & 10 & 6 & 2 & 1 & 7\end{array}\right) \in \mathbf{S}_{10}$;

$g=\left(\begin{array}{llllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 7 & 4 & 6 & 1 & 8 & 3 & 2 & 9 & 5 & 10\end{array}\right) \in \mathbf{S}_{10}$;

$g=(169)(210)(34578) \in \mathbf{S}_{10}$.

Find the matrices of the orthogonal transformations of the plane that carry the rhombus into itself.

Prove that these matrices form, under multiplication, a group $G$ isomorphic to the group $\mathbf{V}_{4}$.

Find the orbits of the action of the group $G$ on the set of vertices of the rhombus and their stabilizers.

the group of rotations of a cube;

the group of rotations of a tetrahedron;

the group of rotations of a dodecahedron.

the group of rotations of an icosahedron is isomorphic to the group $\mathbf{A}_{5}$;

the group of motions of a tetrahedron is isomorphic to $\mathbf{S}_{4}$.

an octahedron;

an icosahedron;

a tetrahedron;

a cube;

a dihedron.

Find the orbits of $G$ in $Y$.

Find the stabilizer $G_{a}$ of a collection $a \in Y$.

Prove that this defines an action of $G$ on $Q$.

Describe the orbits of $G$ in $Q$.

Describe the stabilizer $G_{h}$ of a nonsingular function $h \in Q$.

the stabilizer of the point $O$;

the orbit of the point $O$;

the intersection of the stabilizers of two distinct points of the unit disk.

the map $\sigma_{a}: g H \mapsto a g H$ is a permutation of the set $M$ of all left cosets of the group $G$ modulo the subgroup $H$;

the map $f: a \mapsto \sigma_{a}$ defines an action of the group $G$ on $M$;

$\sigma_{a}$ is the identity permutation if and only if $a$ belongs to the intersection of all subgroups conjugate to $H$ in the group $G$.

$G=\mathbf{Z}_{4}$, $H$ is the trivial subgroup;

$G=\mathbf{D}_{4}$, $H$ is the subgroup consisting of the identity transformation and some reflection of the square.

conjugation defines an action

of the group $G$ on the set $G$;

the stabilizer of a point $m$ (the centralizer of the element $m$) coincides with the set of elements of the group $G$ that commute with $m$.

the permutation (12)(34) in the group $\mathbf{S}_{4}$;

the permutation $(123 \ldots n)$ in the group $\mathbf{S}_{n}$.

$\left(\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right)$;

$\left(\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right)$;

$\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$;

$\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right)$.

all diagonal entries are distinct;

$\lambda_{1}=\ldots =\lambda_{k}=a, \lambda_{k+1}=\ldots =\lambda_{n}=b$ and $a \neq b$.

the centralizer $C_{i j}$ of the elementary matrix $E+E_{i j}$ for $1 \leqslant i \neq j \leqslant n$;

the intersection of the $C_{i j}$ over all $i, j$, where $1 \leqslant i \neq j \leqslant n$;

the conjugacy class containing $E+E_{i j}$.

Prove that any two elementary matrices $E+\alpha E_{i j}$ and $E+\beta E_{p q}$, where $1 \leqslant i \neq j, p \neq q \leqslant n$ and $\alpha , \beta \in F^{*}$, are conjugate.

the centralizer of the operator of rotation by an angle $q \neq k \pi$;

the centralizer of the reflection in the axis $O X$.

$S_{3}$;

$\mathbf{A}_{4}$;

$\mathbf{D}_{4}$.

1;

2;

the permutation (12)(34);

the permutation (124).

$\mathbf{S}_{4}$;

$S_{5}$;

$\mathbf{S}_{6}$;

$\mathbf{D}_{4}$.

if $H$ and $K$ are conjugate subgroups of a finite group and $K \subseteq H$, then $K=H$;

the subgroups

are conjugate in the group $\mathbf{G L}_{2}(\mathbb {R})$, and $K \subset H$.

$G=\mathbf{G L}_{2}(\mathbb {R}), H$ is the subgroup of diagonal matrices;

$G=\mathbf{G L}_{2}(\mathbb {R}), H$ is the subgroup of matrices of the form

$G=\mathbf{S}_{4}, H=\langle (1234)\rangle$.

of the group $\mathbf{Z}_{5}$;

of the group $\mathbf{Z}_{6}$.

Aut $\mathbf{S}_{3} \simeq \mathbf{S}_{3}$, and moreover all automorphisms of the group $\mathbf{S}_{3}$ are inner;

$\mathrm{Aut} \mathbf{V}_{4} \simeq \mathbf{S}_{3}$, and moreover the only inner automorphism of $\mathbf{V}_{4}$ is the identity automorphism.

of the group $\mathbf{Z}_{9}$;

of the group $\mathbf{Z}_{8}$?