The concept of groups

the set of all translations

the set of all translations by nonzero vectors

the set of all rotations about a fixed point

the set of all rotations

the set of all orthogonal transformations

the set of all orthogonal transformations of the second kind

the set of all orthogonal transformations having a common fixed point

the set of all affine transformations

the set of all linear transformations

the set consisting of the identity transformation and the reflection with respect to a given line

the set of rotations of the plane about the center of a regular $n$-gon that carry this $n$-gon onto itself (rotations of a regular $n$-gon)

the set of all similarity transformations?

$x^{ * } = \lambda x, y^{ * } = \lambda y$

$x^{ * } = \lambda y, y^{ * } = \lambda^{ - 1} x, \lambda \neq 0$

$x^{ * } = \lambda x, y^{ * } = y, \lambda \neq 0$

$x^{ * } = x, y^{ * } = \lambda x + y$

$x^{ * } = ax + by, y^{ * } = cx + dy$

$x^{ * } = ax + by, y^{ * } = cx + dy, ad - bc \neq 0$

$x^{ * } = ax - by, y^{ * } = bx + ay, a^{2} + b^{2} \neq 0$

$x^{ * } = r(x\cos \varphi - y\sin \varphi ), \quad y^{ * } = r(x\sin \varphi + y\cos \varphi ), \quad r > 0$

$x^{ * } = a_{1} x + b_{1} y + c_{1}, y^{ * } = a_{2} x + b_{2} y + c_{2}, a_{1} b_{2} - a_{2} b_{1} = 1$?

the set of all real numbers

the set of all nonnegative real numbers

the set of all rational numbers

the set of all integers

the set of all even numbers

the set of all odd numbers

the set of all complex numbers

the set of all purely imaginary complex numbers

the set consisting of the single number 0?

the set of all real numbers

the set of all positive real numbers

the set of all rational numbers

the set of all natural numbers

the set of all nonzero complex numbers

the set of all complex numbers with modulus equal to 1

the set of all nonzero purely imaginary complex numbers

the set of complex $n$-th roots of $1(n$ is a natural number)?

the identity element $e$ is unique

for any element $a$ the inverse element $a^{ - 1}$ is unique

the equality $ax = b$ is equivalent to $x = a^{ - 1} b$, and the equality $xa = b$ is equivalent to $x = ba^{ - 1}$

for any elements $a$ and $b$ the equality $(ab)^{ - 1} = b^{ - 1} a^{ - 1}$ holds.

the group of complex numbers under the operation of addition and the group of translations of the plane under the operation of multiplication (composition)

the group of complex numbers with modulus equal to 1 under the operation of multiplication and the group of rotations of the plane about a fixed point under the operation of multiplication

the group of nonzero real numbers under the operation of multiplication and the group of homotheties with center at a given point under the operation of multiplication (the ratio of the homothety is nonzero)

the group of nonzero complex numbers under the operation of multiplication and the group of transformations of the plane given by the formulas $x^{ * } = ax - by, y^{ * } = bx + ay\left(a^{2} + b^{2} > 0\right)$, under the operation of multiplication

the group of real numbers under the operation of addition and the group of positive real numbers under the operation of multiplication

the group $C_{n}$ of rotations of a regular $n$-gon under the operation of multiplication and the group $U_{n}$ of complex $n$-th roots of 1 under the operation of multiplication.

the group of all integers under addition

the group $n\mathbb {Z}$ of integers that are multiples of a given natural number $n$, under addition

the group $U_{n}$ of complex $n$-th roots of 1 under multiplication

the group $C_{n}$ of rotations of a regular $n$-gon.

the group of all integers under addition

the group $n\mathbb {Z}$ of integers that are multiples of a given natural number $n$, under addition

the group $U_{n}$ of complex $n$-th roots of 1 under multiplication

the group $C_{n}$ of rotations of a regular $n$-gon.

every infinite cyclic group is isomorphic to the group of integers

all finite cyclic groups of the same order are isomorphic to each other.

every subgroup of a cyclic group is itself cyclic

the order of any subgroup of a finite cyclic group is a divisor of the order of the group.

$H$ is a finite set;

all elements of $H$ have finite order.

the group of all orthogonal transformations preserving a given regular $n$-gon (called its symmetry group, also the dihedral group of degree $n, D_{n}$), contains $2n$ transformations

the group $C_{n}$ of rotations of a regular $n$-gon is a normal subgroup of $D_{n}$.

the order of a finite group is divisible by the order of any of its elements

a group of prime order is cyclic.

translations form a normal subgroup of the group of orthogonal transformations of the plane

transformations having a common fixed point form a subgroup of the group of orthogonal transformations of the plane, but it is not normal.

if a finite set of affine transformations of the plane forms a group, then all the transformations in this set have a common fixed point

every finite group of orthogonal transformations of the plane is the symmetry group or the rotation group of some regular polygon.

$G$ is the group of all complex numbers under the operation of addition, $H$ is the subgroup of all real numbers.

$G$ is the group of nonzero complex numbers under the operation of multiplication, $H$ is the subgroup of positive real numbers.

$G$ is the group of nonzero complex numbers under the operation of multiplication, $H$ is the subgroup of numbers with modulus equal to 1.

$G$ is the group of all real numbers under the operation of addition, $H$ is the subgroup of integers.

$G = \mathbb {Z}$ is the group of integers under the operation of addition, $H = n\mathbb {Z}$ is the subgroup of numbers that are multiples of a given natural number $n$.

$G$ is the group of all orthogonal transformations of the plane of the first kind under the operation of multiplication of transformations, $\mathrm{H}$ is the subgroup of translations.

Prove that the set $S_{n}$ of all permutations of degree $n$ is a group under the operation of multiplication of transformations (the symmetric group of degree $n$). Find the order of this group.

Prove that the groups $S_{n}$ are noncommutative for $n \geqslant 3$.

$\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)^{2}$

$\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 1 & 3\end{array}\right) \cdot \left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1\end{array}\right)$

$\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1\end{array}\right) \cdot \left(\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 2\end{array}\right)$

$\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1\end{array}\right) \cdot \left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1\end{array}\right)$ and $\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1\end{array}\right) \cdot \left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1\end{array}\right)$.

$V \subset A_{4}$

$V$ is normal in $S_{4}$

$S_{4} / V \cong S_{3}$.

all subgroups of $S_{3}$

all normal subgroups of $S_{4}$.