Direct products and direct sums. Abelian groups

Prove that the groups $\mathbb {Z}$ and $\mathbb {Q}$ do not decompose as a direct sum of nonzero subgroups.

Do the following groups decompose as a direct product of nontrivial subgroups:

Prove that a finite cyclic group is the direct sum of primary cyclic subgroups.

Prove that the direct sum of cyclic groups $\mathbf{Z}_{m} \oplus \mathbf{Z}_{n}$ is a cyclic group if and only if the greatest common divisor of $m$ and $n$ equals 1.

Decompose as a direct sum the groups:

Prove that the multiplicative group of complex numbers is the direct product of the group of positive real numbers and the group of all complex numbers of modulus 1.

Prove that for $n \geqslant 3$ the multiplicative group of the residue ring $\mathbf{Z}_{2^{n}}$ is the direct product of the subgroup $\left\{ \pm 1\right\}$ and a cyclic group of order $2^{n-2}$.

What is the order of:

Prove that if, in an abelian group, the subgroups $A_{1}, A_{2}, \ldots , A_{k}$ have finite pairwise coprime orders, then their sum is direct.

Let $D$ be a subgroup of the direct product $A \times B$ of groups $A$ and $B$ of coprime orders. Prove that

Let $k$ be the largest order of an element of a finite abelian group $G$. Prove that the order of any element of the group $G$ divides $k$. Is this statement true without the assumption that the group is abelian?

Find all direct decompositions of the group consisting of numbers of the form $\pm 2^{n}$.

Let $A$ be a finite abelian group. Find all direct decompositions of the group $\mathbb {Z} \oplus A$ in which one of the summands is an infinite cyclic group.

Find the conjugacy classes of the group $A \times B$, given the conjugacy classes of the groups $A$ and $B$.

Prove that if the quotient group $A / B$ of an abelian group $A$ by a subgroup $B$ is a free abelian group, then $A=B \oplus C$, where $C$ is a free abelian group.

Prove that a subgroup $A$ of an abelian group $G$ splits off as a direct summand of $G$ if and only if there exists a surjective homomorphism $\pi : G \rightarrow A$ such that $\pi^{2}=\pi$.

Let $\varphi_{1}, \varphi_{2}$ be homomorphisms of groups $A_{1}, A_{2}$ into an abelian group $B$. Prove that there exists a unique homomorphism $\varphi : A_{1} \times A_{2} \rightarrow B$ whose restrictions to $A_{1}$ and $A_{2}$ coincide with $\varphi_{1}$ and $\varphi_{2}$, respectively. Is the commutativity of $B$ essential here?

On the set of homomorphisms of an abelian group $A$ into an abelian group $B$, define an addition operation by the rule

Prove that the homomorphisms $A \rightarrow B$ form an abelian group $\operatorname {Hom}(A, B)$.

Find the groups of homomorphisms:

Prove that $\operatorname {Hom}(\mathbb {Z}, A) \simeq A$.

Let $A$ be an abelian group. Show that all its endomorphisms form a ring with identity $\operatorname {End} A$ under addition and the usual composition of maps.

Prove that the automorphism group of an abelian group coincides with the group of invertible elements of its endomorphism ring.

Find the endomorphism rings of the groups:

Prove that in an abelian group the map $x \rightarrow n x\left(n \in \mathbb {Z}\right)$ is an endomorphism. For which groups is it:

Prove that the endomorphism ring of a free abelian group of rank $n$ is isomorphic to the ring $\mathbf{M}_{n}(\mathbb {Z})$.

Find the automorphism groups of the groups:

Prove that:

Prove that the ring $\operatorname {End}\left(\mathbb {Z} \oplus \mathbf{Z}_{2}\right)$ is infinite and noncommutative.

Prove that the endomorphism ring of a finite abelian group is the direct sum of the endomorphism rings of its primary components.

Prove that a subgroup of a finitely generated abelian group is also finitely generated.

Prove that free abelian groups of ranks $m$ and $n$ are isomorphic if and only if $m=n$.

Let $A, B, C$ be finitely generated abelian groups with $A \oplus C \simeq B \oplus C$. Prove that $A \simeq B$.

Let the order of a finite abelian group $G$ be divisible by the number $m$. Prove that $G$ has a subgroup of order $m$.

Let $A$ and $B$ be finite abelian groups such that, for every natural number $m$, the number of elements of order $m$ in $A$ and in $B$ is the same. Prove that $A \simeq B$.

Let $A$ and $B$ be finitely generated abelian groups such that each of them is isomorphic to a subgroup of the other. Prove that $A \simeq B$.

Prove that a subgroup $B$ of a free abelian group $A$ is free, and that the rank of $B$ does not exceed the rank of $A$.

Using the fundamental theorem on finitely generated abelian groups, find, up to isomorphism, all abelian groups of order:

An abelian group is said to have type $\left(n_{1}, n_{2}, \ldots , n_{k}\right)$ if it is the direct sum of cyclic groups of orders $n_{1}$, $n_{2}, \ldots , n_{k}$.

Does an abelian group of type $(2,16)$ have subgroups of type:

Find the type of the group $\left(\langle a\rangle_{9} \oplus \langle b\rangle_{27}\right) /\langle 3 a+9 b\rangle$.

Are the groups isomorphic:

How many subgroups:

Find all direct decompositions of the groups:

How many elements:

Using the fundamental theorem on finite abelian groups, prove that a finite subgroup of the multiplicative group of a field is cyclic.

Let $F$ be a field whose multiplicative group $F^{*}$ is finitely generated. Prove that the field $F$ is finite.

Prove that a finitely generated subgroup of the multiplicative group of complex numbers decomposes as a direct product of a free abelian group and a finite cyclic group.

Let $A$ be a free abelian group with basis $e_{1}, \ldots , e_{n}$ and $x=m_{1} e_{1}+\ldots +m_{n} e_{n} \in A \backslash 0$, where $m_{i} \in \mathbb {Z}$. Prove that the cyclic group $\langle x\rangle$ is a direct summand of $A$ if and only if the greatest common divisor of the numbers $m_{1}, \ldots , m_{n}$ equals 1.

Let $A$ be a free abelian group with basis $x_{1}, \ldots , x_{n}$. Prove that the elements

form a basis of the group $A$ if and only if $\operatorname {det}\left(a_{i j}\right)= \pm 1$.

Let $A$ be a free abelian group with basis $x_{1}, \ldots , x_{n}$, $B$ its subgroup with generating elements

Prove that the quotient group $A / B$ is finite if and only if $\operatorname {det}\left(a_{i j}\right) \neq 0$, and in that case $\left|A / B\right|=\left|\operatorname {det}\left(a_{i j}\right)\right|$.

Decompose into a direct sum of cyclic groups the quotient group $A / B$, where $A$ is a free abelian group with basis $x_{1}, x_{2}, x_{3}$, $B$ its subgroup generated by $y_{1}, y_{2}, y_{3}$ :

In the quotient group of the free abelian group $A$ with basis $x_{1}, x_{2}, x_{3}$ by the subgroup $B$ generated by $x_{1}+x_{2}+4 x_{3}$ and $2 x_{1}-x_{2}+2 x_{3}$, find the order of the coset $\left(x_{1}+2 x_{3}\right)+B$.

In the quotient group of the free abelian group $A$ with basis $x_{1}, x_{2}, x_{3}$ by the subgroup $B$ generated by $2 x_{1}+x_{2}-50 x_{3}$ and $4 x_{1}+5 x_{2}+60 x_{3}$, find the order of the element $32 x_{1}+31 x_{2}+B$.

Prove that the endomorphism ring of a finite abelian group is commutative if and only if each of its primary components is cyclic.

An additive subgroup $H$ of $n$-dimensional real space $\mathbb {R}^{n}$ is discrete if there exists a neighborhood $U$ of zero such that $U \cap H=0$. Prove that a discrete subgroup of $\mathbb {R}^{n}$ is a free abelian group and its rank does not exceed $n$.

Find all elements of finite order in $\mathbb {R}^{n} / \mathbb {Z}^{n}$.

Let $H=\mathbb {Z}\left[i\right]$ be the subgroup of Gaussian integers in the additive group of the field of complex numbers $\mathbb {C}$. Suppose that $z=x+i y \in \mathbb {C} \backslash H$, where $x, y \in \mathbb {R}^{*}$, and $x y^{-1}$ is irrational. Prove that $\langle z\rangle +H$ is everywhere dense in $\mathbb {C}$.

Let $H$ be a closed additive subgroup of $\mathbb {R}^{n}$. Prove that $H=L \oplus H_{1}$, where $L$ is a subspace of $\mathbb {R}^{n}$ and $H_{1}$ is a discrete subgroup of $\mathbb {R}^{n}$.

Prove that if the order of an element $a$ of an abelian group $A$ is coprime to $n$, then the equation $n x=a$ has a solution in $A$.

An abelian group $A$ is called divisible if the equation $n x=a$ has a solution in it for every $a \in A$ and every integer $n \neq 0$.

Prove that a group is divisible if and only if, for every $a$ and every prime $p$, the equation $p x=a$ has a solution.

Prove that a direct sum is divisible if and only if all the direct summands are divisible.

Prove that the groups $\mathbb {Q}$ and $\mathbf{U}_{p^{\infty }}$ ($p$ a prime number) are divisible.

Prove that a torsion-free group can be given the structure of a vector space over the field $\mathbb {Q}$ if and only if it is divisible.

Let $A$ be a divisible subgroup of a group $G$, and let $B$ be a maximal subgroup of $G$ such that $A \cap B=\left\{ 0\right\}$ (such a subgroup always exists). Prove that $G=A \oplus B$.

Prove that in any abelian group there exists a divisible subgroup such that the quotient group by it has no divisible subgroups.

Let $A$ be a finitely generated abelian group and $B$ a subgroup of $A$. Suppose that $A / B$ is a torsion-free group. Then $A=B \oplus C$, where $C$ is a free abelian group.

Let $A, B$ be free abelian groups and $\varphi : A \rightarrow B$ a group homomorphism. Prove that $\operatorname {Ker} \varphi$ is a direct summand of $A$.

Let $A$ be a free abelian group with basis $e_{1}, \ldots , e_{n}$, $C$ an integer square matrix of size $n$. Denote by $B$ the set of all vectors $x_{1} e_{1}+\ldots +x_{n} e_{n} \in A$ such that

Prove that $B$ is a subgroup of $A$ that is a direct summand of $A$. Conversely, every direct summand of $A$ is given by a system of linear homogeneous integer equations.