Divisible group

In mathematics , a group G is called divisible or divisible if every group element can be divided by every natural number. This means: for every group element and every natural number there is a group element , so that ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle x}$

${\displaystyle g=\underbrace {x\ast \ldots \ast x} _{n{\text{-mal}}}}$

applies. The group link was written with an asterisk . ${\displaystyle \ast }$

If (as is usual with Abelian groups) the link in the group is written as an addition, the defining condition means: For every natural number there is a with ${\displaystyle g\in G}$ ${\displaystyle n}$ ${\displaystyle x\in G}$

${\displaystyle g=x+\ldots +x=n\cdot x}$.

Each group element is therefore divisible by. ${\displaystyle g}$ ${\displaystyle n}$

If the connection is written as a multiplication, as is usual with general groups, the condition means: For every natural number there is a with ${\displaystyle g\in G}$ ${\displaystyle n}$ ${\displaystyle x\in G}$

${\displaystyle g=x\cdot \ldots \cdot x=x^{n}.}$

So there is a -th root of . ${\displaystyle n}$ ${\displaystyle g}$

The background to this is the obvious question: When is a number divisible or divisible by a natural number ? This is generalized to groups. Euclid already described the problem: For which numbers is the equation solvable. Which numbers are multiples of a given natural number. ${\displaystyle y}$ ${\displaystyle n}$ ${\displaystyle y}$ ${\displaystyle y=x+x=2\cdot x}$

${\displaystyle y=n\cdot x=\underbrace {x+\ldots +x} _{n{\text{-mal}}}}$.

At first glance, Euclid deals with a different topic in Book 10 and proves that there is no fraction that solves the equation . For what numbers is the equation ${\displaystyle x^{2}=2}$ ${\displaystyle y}$

${\displaystyle y=\underbrace {x\cdot \ldots \cdot x} _{n{\text{-mal}}}=x^{n}.}$

solvable? If you express these two questions with the help of illustrations, the common background lights up.

• Is , so is the picture ${\displaystyle n\in \mathbb {N} ,n\geq 2}$

${\displaystyle f\colon \mathbb {Z} \ni z\mapsto n\cdot z\in \mathbb {Z} }$not surjective. But the mapping is surjective. ${\displaystyle f\colon \mathbb {Q} \ni q\mapsto n\cdot q\in \mathbb {Q} }$

• The mapping is not surjective. But the mapping is surjective. ${\displaystyle :g\colon \mathbb {Q} \ni q\mapsto q^{n}\in \mathbb {Q} }$ ${\displaystyle f\colon \mathbb {C} \ni z\mapsto z^{n}\in \mathbb {C} }$

This observation suggests to abstract from the whole numbers and the fractions.

Definition of the divisible group

For a group and a natural number , the following statements are equivalent: ${\displaystyle G}$ ${\displaystyle n}$

1. For each there is one with . ${\displaystyle y\in G}$ ${\displaystyle x\in G}$ ${\displaystyle y=x^{n}}$
2. For every homomorphism there is a homomorphism with . Here is the inclusion figure. ${\displaystyle f\colon n\cdot \mathbb {Z} \rightarrow G}$ ${\displaystyle f^{*}\colon \mathbb {Z} \rightarrow G}$ ${\displaystyle f^{*}\circ \iota =f}$ ${\displaystyle \iota \colon n\cdot \mathbb {Z} \hookrightarrow \mathbb {Z} }$

If one of the statements and thus both apply to the group , the group is called divisible by . The group is called divisible if it is divisible by any natural number. In English literature, such groups are called divisible . Sometimes such a group is called divisible . If the group is written additively, the condition is 1 .: . ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle y=n\cdot x}$

Examples

• ${\displaystyle \mathbb {Z} }$with addition as a link, it is not divisible by any natural number . ${\displaystyle >1}$
• The set of real numbers with a terminating decimal fraction becomes a group together with the addition. This is divisible by and , but not by any other prime number. ${\displaystyle 2}$ ${\displaystyle 5}$
• The most important example is the additive group of rational numbers . The element you are looking for is even clear here. ${\displaystyle (\mathbb {Q} ,+)}$
• The additive group of every vector space over the rational numbers is divisible, this is especially true for ${\displaystyle (\mathbb {Q} ^{n},+),(\mathbb {R} ,+),(\mathbb {C} ,+),(\mathbb {R} ^{n},+),(\mathbb {C} ^{n},+)}$
• A group homomorphism maps divisible groups to divisible groups, in particular quotients of divisible groups are divisible: z. B. ${\displaystyle (\mathbb {Q} /\mathbb {Z} ,+),(\mathbb {R} /\mathbb {Z} ,+)}$
• A finite group G is divisible if and only if | G | = 1, because otherwise the exponentiation with n is not surjective
• The examiner group is divisible for every prime number p . ${\displaystyle \left(\bigcup _{k\in \mathbb {N} }\left({\frac {1}{p^{k}}}\mathbb {Z} \right)/\mathbb {Z} ,+\right)\subseteq (\mathbb {Q} /\mathbb {Z} ,+)}$
• the unit group of quaternions is a non-commutative example of a divisible group. ${\displaystyle \mathbb {H} ^{\times }=\left(\mathbb {H} \backslash \{0\},\cdot \right)}$
• Another non-commutative example is the three-dimensional special orthogonal group consisting of the rotations im . ${\displaystyle \operatorname {SO} (3)}$ ${\displaystyle \mathbb {R} ^{3}}$

Divisible Abelian groups

For an Abelian group , the following statements are equivalent. ${\displaystyle (Q,+)}$

1. ${\displaystyle Q}$ is divisible.
2. There is a homomorphism for all subgroups and all homomorphisms , so that is. Here is the inclusion figure. That is, the following diagram is commutative. ${\displaystyle n\mathbb {Z} {\overset {\iota }{\hookrightarrow }}\mathbb {Z} }$ ${\displaystyle f\colon n\mathbb {Z} \rightarrow Q}$ ${\displaystyle f^{*}\colon \mathbb {Z} \rightarrow Q}$ ${\displaystyle f^{*}\circ \iota =f}$ ${\displaystyle \iota \colon n\mathbb {Z} \to \mathbb {Z} ,\ x\mapsto x}$

   ${\displaystyle {\begin{array}{ccc}n\mathbb {Z} &{\overset {\iota }{\hookrightarrow }}&\mathbb {Z} \\\downarrow f&\swarrow &f^{*}\\Q&&\end{array}}}$

3. For every monomorphism and every one there is such that . ${\displaystyle \alpha \colon A\rightarrow B}$ ${\displaystyle f\colon A\rightarrow Q}$ ${\displaystyle f^{*}\colon B\rightarrow Q}$ ${\displaystyle f=f^{*}\circ \alpha }$
4. For all monomorphisms , the mapping is an epimorphism . Where is the set of homomorphisms . ${\displaystyle \alpha \colon A\rightarrow B}$ ${\displaystyle \operatorname {Hom} (\alpha ,Q)\colon \operatorname {Hom} (B,Q)\to \operatorname {Hom} (A,Q),\ f^{*}\mapsto f^{*}\circ \alpha }$ ${\displaystyle \operatorname {Hom} (A,Q)}$ ${\displaystyle A\rightarrow Q}$

Property 2 or 3 means that there is an injective object in the category of Abelian groups . The equivalence of 2nd and 3rd is the Baer criterion (according to Reinhold Baer ). ${\displaystyle Q}$

Direct products of divisible - i.e. injective - Abelian groups are divisible. This applies to every module category. The direct sum of divisible groups is divisible. In general, the direct sum of injective modules is not injective. The epimorphic image of a divisible group is divisible. So is also divisible with. This is a particularly important divisible Abelian group. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} /\mathbb {Z} }$

Injective sheath

There are enough injective groups

${\displaystyle \mathbb {Z} }$is a subgroup of the Abelian group . Each Abelian group can be embedded monomorphically into a divisible Abelian group. There are enough injectives in the category of Abelian groups. This results in: ${\displaystyle \mathbb {Q} }$

For an Abelian group, the following statements are equivalent:

• G is injective.
• For every monomorphism there is a homomorphism with . It is the identity of G. ${\displaystyle \alpha \colon G\rightarrow H}$ ${\displaystyle \beta \colon H\rightarrow G}$ ${\displaystyle \beta \circ \alpha =\mathbf {1} _{G}}$ ${\displaystyle \mathbf {1} _{G}}$

In particular, a divisible group is a direct summand in every parent group .

Injective sheath

${\displaystyle \mathbb {Z} }$is included in the injective group in a special way . If a monomorphism is in any divisible group, then there is a . It is and therefore . Hence it is a monomorphism. is therefore contained in every divisible group that contains, except for isomorphism . is the injective shell of . This is available for every Abelian group G. To clarify this, the large subgroup is defined. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \alpha \colon \mathbb {Z} \rightarrow D}$ ${\displaystyle \alpha ^{*}\colon \mathbb {Q} \rightarrow D\quad {\textrm {mit}}\quad \alpha =\alpha ^{*}\circ \iota }$ ${\displaystyle \operatorname {Kern} (\alpha ^{*})\cap \mathbb {Z} =\{0\}}$ ${\displaystyle \operatorname {Kern} (\alpha ^{*})=\{0\}}$ ${\displaystyle \alpha ^{*}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} }$

Big subgroup

A subgroup is large in G if the only subgroup of G, which with U the cut has. The following statements are therefore equivalent: ${\displaystyle U\hookrightarrow G}$ ${\displaystyle \{0\}}$ ${\displaystyle \{0\}}$

• Every homomorphism with is a monomorphism. ${\displaystyle \alpha \colon G\rightarrow H}$ ${\displaystyle \operatorname {Kern} (\alpha )\cap U=\{0\}}$
• There is one for everyone . ${\displaystyle 0\neq x\in G}$ ${\displaystyle z\in \mathbb {Z} \quad {\textrm {mit}}\quad 0\neq x\cdot z\in U}$

A monomorphism is called essential if is large in H. ${\displaystyle \alpha \colon G\rightarrow H}$ ${\displaystyle \alpha (G)}$

Existence of an injective shell

The following sentence applies:

For every Abelian group G there is a divisible group D and an essential monomorphism . This D is unique except for isomorphism. It's called the injective hull of G and is sometimes referred to as. ${\displaystyle \alpha \colon G\rightarrow D}$ ${\displaystyle D(G)}$

This statement applies to all module categories. Each module over a unitary ring has an injective shell. is the injective shell of . The examiner group for the prime number p is the injective envelope of each group of the species . ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$

Structure set of divisible Abelian groups

Every divisible Abelian group is isomorphic to a (possibly infinite) direct sum of -vector spaces and examiner groups . ${\displaystyle \mathbb {Q} }$

The Abelian group ${\displaystyle \mathbb {Q} /\mathbb {Z} }$

A special Abelian group is . She is a great helper in building the theory of Abelian groups. ${\displaystyle \mathbb {Q} /\mathbb {Z} }$

• ${\displaystyle \mathbb {Q} /\mathbb {Z} }$As an epimorphic image of the divisible group, it is itself divisible and therefore injective. ${\displaystyle \mathbb {Q} }$
• ${\displaystyle \mathbb {Q} /\mathbb {Z} }$is isomorphic to the group of roots of unity in . This is the set of all complex numbers for which there is a natural number n with . ${\displaystyle \mathbb {C} }$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle \quad z^{n}=1}$
• ${\displaystyle \mathbb {Q} /\mathbb {Z} }$contains a copy of each cyclic torsion group . That means: for every natural number there is a monomorphism . ${\displaystyle n}$ ${\displaystyle \alpha \colon \mathbb {Z} /n\mathbb {Z} \rightarrow \mathbb {Q} /\mathbb {Z} }$
• ${\displaystyle \mathbb {Q} /\mathbb {Z} }$thus also contains the injective hull of every simple cyclic group . Where p is a prime number. This is the review group. The endomorphism ring of is isomorphic to the ring of the P-adic numbers . ${\displaystyle D(\mathbb {Z} /p\mathbb {Z} )}$ ${\displaystyle D(\mathbb {Z} /p\mathbb {Z} )}$
• For every Abelian group there is an index set and a monomorphism . It is said to be an injective cogenerator in the category of Abelian groups. ${\displaystyle A}$ ${\displaystyle I}$ ${\displaystyle \alpha \colon A\rightarrow (\mathbb {Q} /\mathbb {Z} )^{I}}$ ${\displaystyle \mathbb {Q} /\mathbb {Z} }$
• The functor not only receives exact sequences , but also discovers them. That means: If a homomorphism is Abelian groups and is an epimorphism, then it is a monomorphism. This results, for example, in the following interesting relationship between divisible and torsion-free Abelian groups: A group is torsion-free if and only if it is divisible. ${\displaystyle \operatorname {Hom} _{\mathbb {Z} }(-,\mathbb {Q} /\mathbb {Z} )}$ ${\displaystyle \alpha \colon A\rightarrow B}$ ${\displaystyle \operatorname {Hom} (\alpha ,\mathbb {Q} /\mathbb {Z} )\colon \operatorname {Hom} (B,\mathbb {Q} /\mathbb {Z} )\ni f\mapsto f\alpha \in \operatorname {Hom} (A,\mathbb {Q} /\mathbb {Z} )}$ ${\displaystyle \alpha }$ ${\displaystyle A}$ ${\displaystyle A^{+}:=\operatorname {Hom} _{\mathbb {Z} }(A,\mathbb {Q} /\mathbb {Z} )}$
• Every finitely generated torsion group is isomorphic to its dual group in this sense . ${\displaystyle A}$ ${\displaystyle A^{+}}$

Individual evidence

1. ↑ Euclid: The Elements. Book I - XIII. Translated from the Greek and edited by Clemens Thaer . 7th, unchanged edition. Scientific Book Society, Darmstadt 1980, ISBN 3-534-01488-X .
2. ^ Friedrich Kasch: Modules and Rings. Teubner, Stuttgart 1977, ISBN 3-519-02211-7 , p. 86.