Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

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Group endomorphisms of simple abelian groups which do not commute by composition.

What is an example of group homomorphisms $f,g: M \to M$ where $M$ is a simple abelian group such that $f\circ g \ne g\circ f$ ?
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36 views

Fixed Point of Automorphism group of a Cyclic group Z2XZ2^2 I need the command on GAP

Dear Mathematics Stack Exchange, I have a problem that how to write a command in GAP the automorphism group of finite abelian group and their fixed points. Let Z_pXZ_p2 be cyclic group where p is ...
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1answer
33 views

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$ i.e. $[-1] \ne [2^k] , \forall k >0$ in $\mathbb Z_n$ ? Is there any ( at least sufficient ) ...
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Subgroups of finite abelian groups.

For every subgroup $H$ of a finite abelian group $G,$ there exists a subgroup $N$ of $G$ such that $G/N \cong H.$ I need to prove this or give a counter example. I am aware of isomorphism theorems ...
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Additivity of trace

Let $A$ be a finitely generated abelian group and $\alpha:A\to A$ be an endomorphism. Since $A=A_{free}\oplus A_{torsion}$, we can induce $\bar \alpha:A_{free}\to A_{free}$, i.e. $\bar\alpha$ is a map ...
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34 views

Order of this group?

It s a stupid question probably but i dont know. It was a little question in a test. The order ( cardinality) of $G= \mathbb{Z}_2 \times \mathbb{Z}_6$. I think it s $12$, the direct product is the ...
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60 views

Which group is isomorphic to?

If I have an abelian group $G$ of order $p^n$, how can I decide if it's isomorphic to $\Bbb{Z}_p \times \Bbb{Z}_p \times\ldots \times \Bbb{Z}_p$ ($n$ times) or to $\Bbb{Z}_{p^2} \times \Bbb{Z}_p \...
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Decomposition of quotient group of lattices

By the Chinese remainder theorem, we know that $\mathbb{Z}_m \cong \prod_{i=1}^l \mathbb{Z}_{p_i^{k_i}}$, where $m=p_1^{k_1} ... p_l^{k_l}$. Now, let $\Lambda = A(\mathbb{Z}^n) \subseteq \mathbb{Z}^...
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Proof that a transitive permutation group (G, X) with G abelian, is sharply regular

As the title states, the question is the following: Let (G, X) be a transitive permutation group, where G is abelian. Show that (G, X) is "sharply regular". First of all I want to notice that in my ...
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Properties of finite abelian group

Let $G$ be a finite abelian group of order $n$ . Then choose the correct statement. a) If d divides n, then there exist a subgroup of $G$ of order $d$ b) If d divides n, then there exist an ...
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Abelian Group Structures

How can I determine all the subgroups of a commutative group, write the Hasse diagram, using Frobenius-Stickelberger Theorem and the isomorphism to $\mathbb{Z}_m$ of a cyclic group? In particular, for ...
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1answer
49 views

Example of an abelian group $G$ with $A \le G$ but no $B \le G$ with $G = A \oplus B$.

I just read that if $G$ is an abelian group with subgroup $A$, then we could not always find a subgroup $B$ such that $G = A \oplus B$. I tried to come up with an example, let $G = \mathbb Z^{\mathbb ...
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Non-abelian group with infinitely many abelian subgroups

I'm looking for a non-abelian group which has infinitely many abelian subgroups. Do you know any examples of such groups?
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Is $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\cong 0$?

Since each 'generator' of $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}$ has the form $km\otimes_{\mathbb{Z}}\bar{a}=k\otimes_{\mathbb{Z}}m\bar{a}=k\otimes_{\mathbb{Z}}0=0$.
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Elementary Abelian p groups

How to show that if any group of class 2 has only two conjugacy class sizes, then its centre and quotient by commutator both are elementary abelian?
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Order of subgroup generated by two cyclic subgroups in $S_6$.

Let $S_6$ be the symmetric group, and $\alpha=(13456)$ and $\beta=(132)$ be its two permutations. How can we find the order of the subgroup generated by $\alpha$ and $\beta$. SOl: $\alpha^5$=...
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Existence of open subgroup extending a smaller one

Let $G$ be an abelian topological group and $H \subseteq G$ a dense subgroup (equipped with the subset topology). Furthermore let $V \subseteq H$ be a subgroup that is open in $H$. Does there exist a ...
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1answer
33 views

Quotient group of free abelian group

Let $A$ be a free abelian group, i.e. $A=\bigoplus_\alpha \mathbb Z$. Also let $B$ be a subgroup of $A$. Prove that $A/B\cong\mathbb Z$ implies $A=B\oplus \mathbb Z$. p.s. Actually this appears in ...
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39 views

adding a relation to a group

We start with a group $G$ such that there is an abelian subgroup $\mathbb{Z}^3(e,f,g) \subset G$. Assume we have $G/\left\{ g = 1 \right\} \cong \mathbb{Z}^2(e,f)$. What can one say about $G$, except ...
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1answer
32 views

Proving that the set of positive integers does not form a group under addition

The set of positive integers under addition has closure because it goes on for infinity and you will always have the element $a + b$. I am also aware that addition is associative but do we include $0$...
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1answer
64 views

Are all complete groups abelian?

Hi: Let $G$ be a group and $G'$ it's commutator subgroup. Then $G > G' > 1$ is a series of normal subgroups of $G$. Suppose $G$ is complete. Then, if I'm not wrong, $Aut(G)$ is the stabilizer of ...
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multiple of a crossed homomorphism from finite group to a divisible one is principal

Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed ...
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Can you prove that modules over a field are vector spaces in a categorical way?

Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category (i.e....
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Isomorphism classes and invariant factors of abelian group

Let $G$ be an abelian group with $ord(G)=3374=2\cdot 7\cdot 241$. Calculate all isomorphism classes with the invariant factors $k_1\ ...k_n$ sucht that $k_i$ divides $k_j$ $(i<j)$. Since $ord(G)=...
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1answer
34 views

When this map $f(x)=x^2$ will be a automorphism on $G$, where $G$ is a commutative group.

I know that for a commutative group $G$ the map $f(x)=x^2$ is a homomorphism from $G$ to $G$. My question : When this map $f$ will be a automorphism on $G$, where $G$ is a commutative group. In other ...
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98 views

If $a^3=1$, is $G$ abelian?

If $G$ is a group that satisfies $a^3=1$ for every $a\in G$, then is $G$ abelian? This is an exercise I found in Jacobson's Basic Algebra. It is analogous to the question: If $G$ is a group that ...
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3answers
98 views

Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Since when are free abelian groups constructed w.r.t maps? Isn't the set $S$ all that matters? I don't understand what ...
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1answer
37 views

Let $G$ be an abelian group can we always construct a quasi projective variety $X$ such that Cl$(X)=G$

Recall that for a quasi projective variety $X$ one can define the Divisor Class Group denoted by Cl$(X)$ Let $G$ be an abelian group. Can we always construct a quasi projective variety $X$ such ...
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What is the range of this function?

What is the range of $h$? $f(x)=4x+1$ $g(x)=(x-1)/3$ Let $h=\{f^n(g^m(1)):n,m\in\mathbb{N}\geq0\}$ What is the range of $h$? Show that $(2\mathbb{N}-1)\subset H$. ... okay I've done a bit more: ...
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Show that open interval $(-1,1)$ is isomorphic to $(\mathbb{R},+)$

Define group structure on $G=(-1,1)$ by $$a*b=\frac{a+b}{1+ab}$$ for any $a,b\in G$. Show that $G$ is isomorphic to $\mathbb{R}$ under addition. I've tried the obvious maps $f:G\rightarrow \mathbb{R}$...
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Does an abelian subgroup inject into the abelianisation of the whole group? [closed]

If $H <G $ are groups and H is abelian, do we get an injection from H into $G/[G,G] $?
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$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]|$ is even

Let $G$ be a finite group. My problem is: $$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]| \mbox{ is even}.$$ I know that $\hom(G, \Bbb Z/2)=\hom(G^{ab}, \Bbb Z/2)$, where $G^{ab}:=G/[G,G]$. If $|\hom(G^{ab}...
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Are these conditions sufficient for a $\mathbb{Z}$-module to be free?

There exist modules over the integers that, like $\mathbb{Q}$, manage to be torsion-free without being free. Ergo, its probably worth looking for conditions $P$ such that "$P$ + torsion-free" is ...
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1answer
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Why if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$ [duplicate]

Prove that if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$. I have proved the converse, but here there is something I am missing. Hints instead of full answers are appreciated. Thanks.
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1answer
51 views

Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$

I want to determine how many subgroups does the additive group $G:=\Bbb Z_{14} \oplus \Bbb Z_{6}$ have? There are many related posts in our site, for instances: here and there. However, it ...
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2answers
75 views

All subgroups normal $\implies$ abelian group

This is , I think an easy problem just that I am not getting the catch of it. How to show whether or not the statement is true? All subgroups of a group are normal$\implies$ the group is an abelian ...
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1answer
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Subgroup with order =LCM of two subgroups

The following is a question that was asked by my teacher as a ponder-upon question:to which unfortunately I have not been able to put a single forward step. If an abelian group has subgroups of order ...
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3answers
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Abelian finite group [duplicate]

This is a (maybe be simple) problem from Group Theory, but being a beginner, I am unable to take even a first step forward. Let $G$ be a finite group whose order is not divisible by $3$.Suppose that $...
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Showing that the primary component $G_p$ is a subgroup of $G$

For a finite abelian group $G$ and a prime number $p$ with $p \mid |G|$, we define $G_p$ as the subset of $G$ that contains all elements of $G$ with order $p^k$ for a $k \in \mathbb{N}_0$. We call $...
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Prove that G is an Abelian Group

Let G = {x in Q : 0≤x<1}. Define the operation on G: a•b = a+b if 0≤a+b<1, a+b-1 if a+b≥1 Prove that (G,*) is an Abelian group. Attempt: (commutativity was easy). For associativity I got a+b≥1 ...
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Groups and queues and stacks

As I review my elementary CS material to prepare for an interview I cannot help but think that I missed a key connection when studying this prior: I think I missed the relationship between operations ...
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1answer
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Prove that the group of the rational points on the conic $u^2-Av^2=1$ is not finitely generated.

This is an exercise from Rational Points on Elliptic Curves by Silverman. Let $H$ be the conic $u^2-Av^2=1$ where $\sqrt{A}\notin \mathbb{Q}$. If $(u_1,v_1), (u_2,v_2)$ are two points in $H(\...
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A Group G is abelian $\Leftrightarrow$ $ Inn(G)$ is a normal subgroup of Sym(G)

First of all I don´t think that this question is answered here If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$ because in my opinion ...
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Criterion for a group $G$ to be abelian using $Inn(G)$ [duplicate]

Let $(G, *)$ be a group. Let $\mathcal S(G)=\{f:G\to G\space|f\space is\space bijective\}$ be the symmetric group and $Inn(G)=\{\kappa_{a}\space|\space a\in G\}$ the inner automorphisms. Now I want to ...
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1answer
41 views

Fundamental group of a tree?

Find the fundamental group of the space $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$. $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$ where $T$ is a graph $T$ is the graph made of $3$ edges with a ...
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15 views

Conjugacy Class Equation for $\mathbb{Z_{25}}$

I'm suposed to find the conjugacy class equation for $\mathbb{Z_{25}}$. Since $\mathbb{Z_{25}}$ is Abelian, that means that $gxg^{-1}=xgg^{-1}=x$ so $Z(\mathbb{Z_{25}})=\mathbb{Z_{25}}$ and every ...
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1answer
35 views

Homomorphism $f:M \to M$ such that $\mathrm{im}(f) = L$ and $f \circ f = f$ where $L \le M$ and $M$ abelian

Let $M$ be an abelian group. For any subgroup $L$ of $M$, can you find a homomorphism $f:M \to M$ such that $\mathrm{im}(f) = L$ and $f \circ f = f$? This was an attempt to find an idempotent ...
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Does $\phi: A \otimes \mathbb{Q} \to B \otimes \mathbb{Q}$ surj. imply that for $b \in B$, $b = n \phi(a)$ for some $n \in \mathbb{Z}$, $a \in A$?

Let $A$ and $B$ be abelian groups. Suppose that we have a morphism $\phi: A \to B$ and that $\phi \otimes_\mathbb{Z} \mathbb{Q}: A \otimes_\mathbb{Z} \mathbb{Q} \to B \otimes_\mathbb{Z} \mathbb{Q}$ is ...
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44 views

Why abelian groups instead of modules in Algebraic Topology

I am studying Algebraic Topology, homology and cohomology to be concrete. I am reading\working through Hatcher, Rotman, Harper and sometimes I combine them with other books when none of them give a ...
2
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1answer
32 views

Regarding those $\mathbb{Z}$-modules whose every finite subset generates a finite submodule.

Let $X$ denote a $\mathbb{Z}$-module (aka an abelian group). Then $X$ may or may not satisfy: $(*)$ for all finite sets $F \subseteq X$, the module generated by $F$ is finite. This properly ...