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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2
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1answer
53 views

How to show that the centralizer of commutator subgroup is nilpotent?

How to show that $C_G(G')$ (the centralizer of commutator subgroup $G$) is nilpotent? ($G'$ is the commutator of group $G$.)
5
votes
1answer
87 views

Are there proper $1/2$ convex subsets of finite groups?

According to this paper by N. H. Bingham and A. J. Ostaszewski, A subset $H$ of a group $G$ is said to be $1/2$ convex (or mid-point convex) if for each $x,y\in H$, there exist a unique element ...
2
votes
4answers
88 views

Finite groups where $x^2 = e$ has order $2^n$

Prove that a finite group $G$, where $x^2 = e$ $\forall x \in G$, has order $2^n$ for some $n \in \mathbb{N}$. I already know that every group with this property is abelian, but I don't see the ...
1
vote
0answers
20 views

Generated subgroups in an $p$-primary abelian group

I am having trouble understanding the extra assumptions given in this problem: Assume $\{y_1,...,y_t\}\subset G$ where $G$ is an abelian $p$-primary group,with ...
3
votes
1answer
65 views

$G' = [A, B]$ if $G = AB$, where $A,B$ are abelian.

Suppose that $G = AB$, where $A$ and $B$ are abelian subgroups. Show that $G' = [A, B]$. Showing that $[A,B] \subseteq G'$ is simple enough, but how do I show reverse inclusion?
0
votes
2answers
43 views

$G$ is a finite group, prove that $G$ is abelian when $G/C(G)$ is cyclic. [duplicate]

Is it true that when $G/C(G)$ is cyclic, then $G$ is abelian? The original question is Prove that if $G$ is a finite non-abelian group, then $4|C(G)|\le G$. Proof from the answer book: suppose ...
3
votes
2answers
37 views

Prove that $o(a)=o(gag^{-1})$

Let $G$ be a group and $a\in G$. Prove that $o(a)=o(gag^{-1})$ for every element of order $2$ in $G$. If a be the only element of order $2$ in $G$ deduce that a commutes with every element of $G$ ...
-1
votes
2answers
115 views

Prove if any group has order 9, then it is abelian. [closed]

I not getting how to start with this.I have completed Lagrange theorem.
3
votes
1answer
47 views

Exactness of the tensor product

considering the tensor products of abelian groups: could you tell me if (and why?) the following is true? For any free abelian group $A$ the functor $A\otimes (-)$ is exact. Thanks! [I extracted ...
-2
votes
1answer
52 views

How to tell if this set is a group? [closed]

How to tell if this set is a group? $(G,\star)$ where $G = \{(a,b)|a\in\mathbb R$ and $b\in\mathbb Q^*\}$ and $\star$ is defined by $(a,b)\star(c,d) = (a+c, bd)$ for all $(a,b),(c,d)\in G$ I have ...
1
vote
1answer
43 views

Decompose $\mathbb {Z}_{pq}^* / (\mathbb {Z}_{pq}^*)^2$ into a direct sum of cyclic groups.

Let $\mathbb {Z}_{pq}^*$ be the set of all units of $\mathbb {Z}_{pq}$ and $(\mathbb {Z}_{pq}^*)^2 = \{ a^2 \mid a \in \mathbb {Z}_{pq} \}$. ($p, q$ are distinct odd primes.) Decompose $\mathbb ...
0
votes
0answers
39 views

Proof of cyclic groups

So I have this problem: Prove that $\mathbb Z_8 \times \mathbb Z_9$ is a cyclic group, which I solved this way but I'm not sure that the solving is correct or complete: $1) $ $\mathbb Z_m \times ...
5
votes
0answers
92 views

Schröder-Bernstein for abelian groups with direct summands

What is a simple example of two abelian groups $A,B$ which are isomorphic to direct summands of each other (that is, $A \cong B + C$ and $B \cong A + D$ for some abelian groups $C,D$), but which are ...
0
votes
1answer
27 views

CA-group [abelian centralizer group]

I am searching for all information about CA-groups [abelian centralizer group] and i just found a German book [ Huppert ] and Nilpotent Centralizer group of Suzuki in 44 pages. I need more English ...
4
votes
2answers
154 views

Prove that the group $G$ is abelian if $a^2 b^2 = b^2 a^2$ and $a^3 b^3 = b^3 a^3$

In a Group $G$, $a^2b^2=b^2a^2$ and $a^3b^3=b^3a^3$ holds, $\forall a,b\in G$. Prove that the group $G$ is abelian. My approach was the following: Let $a,b\in G$ Then, $a^2b^2=b^2a^2$ and ...
3
votes
0answers
31 views

A question on direct summand subgroups

Let $G$ be an abelian group such that $G$ has no nontrivial direct summand subgroup. Is there a characterization of $G$?
4
votes
1answer
80 views

If $G$ acts on $A$ faithfully/freely/transitively, what can we say about its action on $\hat A$?

Let $A$ be a finite abelian group and let $G\subset \operatorname{Aut}A$, with $G$ acting from the left. $G$ acts in a natural way (on the left) on $A$'s character group $\hat A$ by $g\chi = \chi\circ ...
0
votes
2answers
78 views

Subgroups of finitely generated abelian groups [closed]

Let $G\cong \mathbb{Z}_{{p_1}^{k_1}}\oplus\mathbb{Z}_{{p_2}^{k_2}}\oplus \dots\oplus\mathbb{Z}_{{p_n}^{k_n}}$. Let $H$ be a subgroup of $G$. Does $H$ necessarily have the form $H\cong ...
0
votes
0answers
39 views

Is this true for quotients of finitely generated abelian groups?

Let $G\cong \mathbb{Z}_{{p_1}^{k_1}}\oplus\mathbb{Z}_{{p_2}^{k_2}}\oplus\dots\oplus\mathbb{Z}_{{p_n}^{k_n}}$, where the $p_i$'s are primes. Let $H\cong ...
1
vote
5answers
87 views

$\mathbb{Z}_6/\mathbb{Z}_2$ isomorphic to $\mathbb{Z}_3$?

Recently in class my teacher mentioned that the quotient group $\mathbb{Z}_6/\mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3$. May I ask why is this so? Also, what do elements in ...
1
vote
1answer
77 views

For What Abelian groups $A$ Does There Exist a Short Exact Seqeuence $0\to \mathbb{Z}/p^m\mathbb{Z}\to A\to \mathbb{Z}/p^n\mathbb{Z}\to 0$.

$\newcommand{\Z}{\mathbb Z}$ Question: Let $m$ and $n$ be positive integers. What are all the abelian groups $A$ such that there is a short exact sequence $0\to \mathbb Z/p^m\mathbb Z\to A\to ...
1
vote
2answers
48 views

Homomorphisms from $\mathbb{Z}_p$ to $\mathbb{Z}_3$

For which odd values of $p$ can we find a non-trivial homomorphism from $\mathbb{Z}_p$ to $\mathbb{Z}_3$ ? Is there any method to find those homeomorphisms explicitly? I have no any idea to handle ...
0
votes
0answers
9 views

Property of locally cyclic groups

I am having difficulty proving that: A group $G$ is locally cyclic if and only if $G$ is isomorphic to a subgroup of $\mathbb{Q}$ or $\mathbb{Q/Z}$. Is there any easy way to prove it? Thanks.
0
votes
0answers
20 views

Automorphism group of a locally cyclic group

I am having difficulty proving that: The automorphism group of a locally cyclic group is commutative. Is there any easy way to prove it? Thanks.
1
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0answers
21 views

Folner sequence of $\mathbb{Z}[\frac{1}{2}]$

Consider $\mathbb{Z}[\frac{1}{2}]$ consisting of rational numbers of the form $k2^l$ with $k,l\in\mathbb{Z}$. Under addition and discrete topology $\mathbb{Z}[\frac{1}{2}]$ is a discrete abelian ...
-1
votes
1answer
29 views

Abelian-by-(finite abelian) [closed]

hope you all doing fine. I have a question. Is it true that a abelian-by-(finite abelian) group is also (finite abelian)-by-abelian? Thanks.
0
votes
0answers
40 views

Finitely generated abelian groups and finite index subgroups

I want a proof or reference for the following fact: "Let $G$ be a finitely generated abelian group and let $\phi:G\to G$ be an injective homomorphism. Then the index $[G:\phi(G)]$ is finite." I ...
0
votes
1answer
51 views

About a proof regarding a property of groups of order $pq$ where $p$ and $q$ are primes

I'm studying right now Automorphisms in Dummit & Foote's Abstract Algebra (Section 4.4). In pages 135-136, the following example is given: and here's Proposition 16 muntionned in the ...
1
vote
1answer
33 views

A questions on the groups by a copy of $\Bbb Z$

Let $G$ be an abelian group and $H$ a subgroup of $G$ such that $G/H$ contains a copy of $\Bbb Z$. Is this true that $G$ contains a copy of $\Bbb Z$? ($\Bbb Z$ is the group of integer numbers)
1
vote
1answer
39 views

If $G_3$ is finitely generated abelian group then there is a short exact sequence with $G_2$ and $G_1$ free groups?

Let $G_i$ be abelian Groups. A exact sequence of the form $ 0 \to G_1 \to G_2 \to G_3 \to 0$ is called a short exact sequence. Is the following statement true? If $G_3$ is finitely generated then ...
1
vote
1answer
99 views

Fundamental Theorem of Abelian Groups

From fundamental theorem of finite abelian groups I can say any finite abelian group $G$ is isomorphic to direct sum of cyclic groups i.e, $G\cong Z_{{p_1}^{i_1}}\oplus Z_{{p_2}^{i_2}}\oplus ...
4
votes
1answer
60 views

Abelian group and their subgroups

Is it true that If an abelian group has subgroups of order m and n respectively then it has a subgroup whose order is the least common multiple of m and n? If it is then can anyone explain it with a ...
4
votes
0answers
60 views

Generating Sets for Subgroups of $(\Bbb Z^n,+)$.

The question Finite Generated Abelian Torsion Free Group is a Free Abelian Group led me to conjecture and prove an interesting thing about generating sets for $\Bbb Z^n$ and certain subgroups. If ...
2
votes
0answers
41 views

Finding a property for $G/Z(G)$ where $G$ is a nonabelian group

If $G$ is non-abelian group and $Z(G)$ is it's center, what is the least property for $G$ such that $\frac{G}{Z(G)}$ is abelian?
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0answers
6 views

Is there a characterization for discrete abelian torsion-free and reduced groups?

Is there a characterization for discrete abelian torsion-free and reduced groups? A group is called reduced if it contains no nontrivial divisible subgroups.
0
votes
1answer
43 views

discrete subgroups of euclidean space

I'm trying to prove this proposition: http://groupprops.subwiki.org/wiki/Every_discrete_subgroup_of_Euclidean_space_is_free_Abelian_on_a_linearly_independent_set That every discrete subgroup of ...
1
vote
2answers
59 views

If $f\otimes_\mathbb{Z}\mathbb{Z}/(p)\colon M\otimes_{\mathbb{Z}}\mathbb{Z}/(p)\to N\otimes_\mathbb{Z} \mathbb{Z}/(p)$ is onto for all $p$, $f$ onto?

This lemma is used in a theorem I'm reading, with no proof. Suppose $f\colon M\to N$ is a morphism of free, finitely generated $\mathbb{Z}$-modules. Then if $f\otimes_\mathbb{Z}\mathbb{Z}/(p)$ is ...
2
votes
1answer
24 views

setting abelian group in non-abelian group

Is it right to say for every (finite) abelian group $H$ there is non-abelian group $G$ such that $Z(G)=H$, where $Z(G)$ is the center of $G$?
2
votes
1answer
50 views

A characterization of an abelian group

Let $G$ be an abelian group. Is there a characterization of $G$ whenever every subgroup of $G$ is a direct summand of $G$?
2
votes
1answer
104 views

A group is abelian if and only if the center of the group is all the group

Isn't it the same to say that a group is abelian, and that the center of the group is all the group? I have an exercise to prove that this is true, and it's exactly one stroke for each direction of ...
1
vote
2answers
76 views

What is the rank of $\mathbb{Q}$ over $\mathbb{Z}$?

What is $\operatorname{rank}_{\mathbb{Z}} \mathbb{Q}$? I think it is $\aleph_0$, but cannot figure out how the basis would look like. Thank you in advance. (I faced this when proving ...
1
vote
1answer
16 views

$A/A^{p}\cong A_{p}$ for finite abelian (additive) gp. and prime $p$.

Let $A$ be a finite abelian (additive) gp. and $p$ be a prime. I want to show $A/A^{p}\cong A_{p}$ where $A^{p}:=\left\{pa:a\in A\right\}$ and $A_{p}:=\left\{a\in A:pa=0\right\}$.(I want to show ...
0
votes
0answers
19 views

What are the character functions of $\mathbb{Z}_N \times \mathbb{Z}_N$ ?

$\mathbb{Z}_N \times \mathbb{Z}_N$ is an Abelian group which I can think of to consist of all tuples of the form $(\omega ^a, \omega^b)$ where $0 \leq a,b \leq (N-1)$ and $\omega = e^{ \frac{2 \pi i ...
2
votes
2answers
69 views
2
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2answers
136 views

How to prove the group $G$ is abelian?

Question: Assume $G$ is a group of order $pq$, where $p$ and $q$ are primes (not necessarily distinct) with $p\leqslant q$. If $p$ does not divide $q-1$, then $G$ is Abelian. I know that if the ...
6
votes
2answers
94 views

Cubic Planar Graphs have $2^m-1$ Hamilton Cycles, contradicting Bosak…

I looked at the symmetric difference of hamilton cycle (HC) in cubic planar graphs and found that, together with the empty graph, they build a subgroup of the abelian group $\Omega$ of symmetric ...
2
votes
1answer
36 views

A question on finite abelian groups

Let $m_1,\dots,m_k$ be positive integers. Are there positive integers $d_1,\dots,d_k$ such that $d_i|d_{i+1}$ and $$ \oplus_{i=1}^k \mathbb{Z}/m_i\mathbb{Z}\cong \oplus_{i=1}^k ...
1
vote
1answer
36 views

Definition of quasi-cyclic and full rational groups

In Unit Groups of Classical Rings by Karpilovsky, p.96, we can see this theorem: Let $G$ be a divisible abelian group. Then $G$ is a direct product quasi-cyclic and full rational groups. I want ...
0
votes
1answer
33 views

Exsitence of element of a certain order in an infinite abelian group

I came up with the following question reading this(Finite Abelian Groups question). Let $G$ be an abelian group. Suppose there is an integer $n \ge 1$ such that $nG = 0$. Let $m$ be the smallest ...
0
votes
1answer
36 views

Quotient group of free groups

Let $G=\langle g_1,\ldots,g_k\rangle$ be a free abelian group generated with $g_1,\ldots,g_k$ and let $H=\langle g_{r+1},\ldots,g_k\rangle$ be a free abelian subgroup of $G$. Is it then the case that ...