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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Relation of Smith normal form to basis of subgroup

Let $A$ be a finite abelian group of rank $2$. Let $\left\{ e_{1},e_{2}\right\}$ be a basis of $A$ and let $C=\left\langle 2e_{1}+3e_{2},2e_{1}+6e_{2}\right\rangle $ be a subgroup. (a) Find ...
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1answer
24 views

Exercise 6.79 from Rotman's Advanced Modern Algebra

If $G$ is a nonzero abelian group show that $$\operatorname{Hom}_{\Bbb Z}(G,\frac{\Bbb Q}{\Bbb Z}) \neq \{0\}.$$
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1answer
28 views

Group of units of a non-integral quotient ring

I would to like to know which product of cyclic groups the group $A^\times$ of units of the quotient ring $$ A = \mathbb F_5[X] / ((X^2-2)^2) $$ is isomorphic to. I know that $A$ is not an integral ...
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1answer
34 views

Proving that a group is abelian [closed]

Suppose we have a group $G$ with $|G| = 10$. How do I prove that if its center $Z$ is nontrivial, then $G$ is abelian? Thanks.
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2answers
72 views

Isomorphism and $(\mathbb Q,+)$

Prove that $(\mathbb Q,+)$ is not isomorphic to $(H,+) \neq (\mathbb Q,+)$, a proper subgroup of $(\mathbb Q,+)$. $\mathbb Q$ is the rationals. I thought about taking the contradiction ...
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1answer
33 views

Open mapping theorem for normed abelian groups

A norm on an abelian group is a function valued in $\mathbb{R}_{\geq 0}$ which satisfies $|x|=0 \Leftrightarrow x=0$, $|{-}x|=|x|$, and $|x+y| \leq |x|+|y|$, not necessarily $|z x| = |z| |x|$ for ...
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2answers
47 views

Is there a non abelian group that characterize a one dimensional lattice structure?

Of all groups that characterize a one dimensional lattice structure (symmetry operations including translation, $C_2$, mirror plane, inversion point), is there a non abelian one? Moreover, Can it have ...
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1answer
52 views

Abelian Group G and it's order (Let G be an abelian group, and let a∈G)

I already asked this question. However, it was closed off. I now have inserted by attempt at the solution: Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, let $G[n:a] := \{x\in G:x^n ...
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0answers
34 views

Abelian Groups and orders

I came across this question, and was wondering what the notation G[n : a] means. Is x the generator? Any comments/explanations will be of great help. The question I found was: Let $G$ be an abelian ...
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2answers
115 views

Does there exist an $n$ such that all groups of order $n$ are Abelian?

I know that all groups of order $\leq$ 5 are Abelian and all groups of prime order are Abelian. Are there any other examples? If so is there something special about the orders of these groups?
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2answers
16 views

Struggling to understand example of Ideal which is not finitely generated

I'm working through an algebraic number theory book, but I can't understand the example shown below: I follow the example up till it assumes that $\frac{p_1}{q_1},...,\frac{p_n}{q_n}$ are the ...
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3answers
81 views

Two term free resolution of an abelian group.

This is probably a very easy question but I think I am missing some background regarding free abelian groups to answer it for myself. In Hatcher's Algebraic Topology, the idea of a free resolution is ...
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42 views

Difficulty understanding subgroups for certain simple groups

Firstly, I am studying at the high-school level so please excuse my lack of understanding of these concepts. Consider the group $\Bbb C_{3v}$ of symmetries of an equilateral triangular lamina. It ...
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0answers
49 views

Extending a homomorphism from a subgroup to whole group where the target is not a divisible group

I was reading this post of stack exchange. So in the question if the circle group is replaced by $\mu_{p-1}$ which is the group of $(p-1)^{th}$ root of unity and if the group $G/H$ is assumed to a ...
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1answer
42 views

Number of group homomorphism from $Z_8$ ⊕ $Z_2$ to $Z_4$ ⊕ $Z_4$

I know that there does not exist an isomorphism from $Z_8$ ⊕ $Z_2$ to $Z_4$ ⊕ $Z_4$ as there exist an element of order 8 in $Z_8$ ⊕ $Z_2$ and no element of order 8 in $Z_4$ ⊕ $Z_4$. But what about ...
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1answer
49 views

Proving that a homomorphism is abelian [duplicate]

The question states... Let $\phi: G \to G$ be a homomorphism with the map $\phi(g)=g^2$. Prove $\phi$ is abelian. So far I have: Let $g$ and $h$ be in $G$. Then $\phi(g)=g^2$ and $\phi(h)=h^2$. ...
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33 views

Order of elements in certain generating sets of non-abelian groups !!!

The following example is just to clarify the idea. Example: The dihedral group has the following presentation $$D_{2i}=\left<s,r/s^2=r^i=e,sr=r^{-1}s \right>$$ Let $S_1=\{s,r\}$, ...
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1answer
54 views

Subgroups of abelian-by-finite groups

I am trying to prove that a subgroup of a abelian-by-finite group is also abelian-by-finite. I am not sure if I can use the same procedure that is used for a subgroup of a ...
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1answer
44 views

Divisible group $G\neq 0$ is not free

How do I show that a divisible group $G\neq 0$ is not free? I know that divisible means that for all elements $g$ in an abelian group $G$ and $n\in\mathbb{N}$ there is an element $a\in G$ such that ...
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0answers
54 views

Structure theorems for infinitely generated Abelian groups

The classification theorem for finitely generated Abelian groups is well known and plays big role in mathematics. Are there any structure theorems about infinitely generated Abelian groups known?
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1answer
69 views

Isomorphism between $A$ and $B$ if $A/kA$ and $B/kB$ have same order? [closed]

Let $A$ and $B$ be finitely generated abelian groups, where for all $k\geq1$ the orders of $A/kA$ and $B/kB$ are equal. Are $A$ and $B$ necessarily isomorphic? Why (not)?
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1answer
106 views

Surjective endomorphism of abelian group is isomorphism

Let $A$ be a finitely generated abelian group and $f:A\rightarrow A$ a surjective homomorphism. How do I prove that $f$ is an isomorphism? And if $f$ were injective instead of surjective would the ...
3
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1answer
105 views

Finding an explicit isomorphism from $\mathbb{Z}^{4}/H$ to $\mathbb{Z} \oplus \mathbb{Z}/18\mathbb{Z}$

There was a past qualifying exam problem, I was having trouble with, it is stated below as follows: In the group $G= \mathbb{Z} \times \mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}=\mathbb{Z}^{4}$, ...
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37 views

Proving property of cyclic groups

A user asked the following question. It was closed as off-topic, or rather as missing context, but it seems the context close reason doesn't exist, so off-topic was chosen. Here it is: I am having ...
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1answer
52 views

Exercises Involving Torsion and Abelian Groups

I am working on the following group-theory exercises but I'm a little confused by how to begin proving them. Let $G$ be a group. Call $g \in G$ a $torsion \ element$ if $g$ has finite order $g^k = e$ ...
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1answer
54 views

o(a)=m ,o (b) =n , ab=ba then o(ab)=lcm(m,n). What happens when b is the inverse of a?

Let $G$ be a group and let $a,b \in G$ s.t $O(a)=m$ and $O(b)=n$ and $ab=ba$. Then $O(ab)=lcm( m,n)$. My attempt: since $ab=ba$ then $HK=KH$ $ |HK|=O(H)O(K)/O(H \cup K)$ $l=(mn)/O(H \cap K)$ ...
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1answer
100 views

Why is abelianness such a precious property?

My abstract algebra teacher said the other day that constructions like ideals and cosets and normal subgroups are "trying to capture a little bit of abelianness." He has used phrases like "magic ...
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1answer
97 views

Orders of the elements in $\mathbb{Z}/8\mathbb{Z}$

I know that the order of an element $a$ in a group is the smallest positive integer $n$ such that $a^n = 1$. You know $\mathbb{Z}/8\mathbb{Z} = \{\overline{0}, \overline{1}, \dotsc, ...
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66 views

Which lines in $\mathbb R^2$ define a subgroup?

Which lines in $\mathbb R^2$ define a subgroup? I know that the line $y=x$ in $\mathbb{R}^2$ gives a subgroup, but I can't figure out the other ones.
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34 views

Find two groups that give a cyclic decomposition of $G=\mathbb{Z}_4 \oplus \mathbb{Z}_2$

Let $G=\mathbb{Z}_4 \oplus \mathbb{Z}_2$. Find two distinct pairs of subgroups of G such that each pair gives a cyclic decomposition of G with no subgroup of the second pair equal to either subgroup ...
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1answer
59 views

Is this construction algebraically closed?

On the tetration forum Tommy1729 proposed a new kind of number : http://math.eretrandre.org/tetrationforum/showthread.php?tid=1036 Too avoid deletion or changes of that post , I copy it here : ...
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2answers
67 views

Commutative free products

Do there exist any non-trivial groups such that their free product is commutative? That is, if $G, H$ are non-trivial groups is $G*H$ ever commutative? My thinking is no but I can't really formulate ...
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0answers
51 views

Questions related to semidirect-product of Klein four group?

I have four questions related to Klein four group. and I know the answer two of them. ( the first two) and I want to know answer Does $V_4$ has an automorphism of order 6? What are the orders of ...
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1answer
31 views

Algebraic structures [closed]

I can't wrap my head around this area in mathematics. What is a group, a, semigroup, what is a field, a ring, an abelian group? I read all sorts of texts, but it's so abstract. I can't solve problems ...
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1answer
42 views

How to construct an abelian permutation group?

I am studying binary operations and permutation groups. One of the exercices leaves me a bit perplex, that is: "Construct a permutation group which is Abelian(commutative)" From what I understand, a ...
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1answer
50 views

$\text{Aut}(G) \cong \text{Out}(G)$ when $G$ is abelian

I would appreciate if you could please evaluate my proof and point out any mistakes I made. Proof: Define a homomorphism $\Phi: \text{Aut}(G)\to \text{Out}(G)$, such that all elements of ...
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2answers
199 views

$Aut(G)$ is cyclic $\implies G$ is abelian

I would appreciate if you could please express your opinion about my proof. I'm not yet very good with automorphisms, so I'm trying to make sure my proofs are OK. Proof: Since $Aut(G)$ is cyclic, ...
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0answers
18 views

Finite abelian group of orthogonal matrices

Let $G$ be a finite abelian subgroup of $O_{2n+1}(\mathbb R)$. Suppose that $G\not\subset \{-I_n,I_n\}$. There exists $\varepsilon \in \{-1,1\}$ and $M\in G$ such that $0<\dim \ker (M-\varepsilon ...
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2answers
38 views

Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1 }$.

Let $(G, *)$ be a group and let $\{g,h\}$ be a subset of $G$. Prove that $(g*h)^{-1}=h^{-1}*g^{-1}$. I know that I should show that $X*Y=Y*X=e$. But I don't know how to calculate it.
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1answer
20 views

Subquotient of abelian group equivalent to “quotientsub” of abelian group?

Suppose we have an abelian group $G$. Suppose also that we have a "subquotient" $H$, which is a subgroup of a quotient group of $G$. If $H$ can be constructed in this way, when is it also true that ...
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0answers
28 views

Group G with $\left(\forall g \in G: g^2=1_G\right) \implies \text{G is abelian}$ [duplicate]

I have the following task: Be $G$ a group with $\forall g \in G: g^2=1_G$ Prove that $G$ is abelian. I proved it this way: $\forall g \in G: g^2=1_G$ implies $\forall g \in G: g=g^{-1}$ ...
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1answer
133 views

Let G be an abelian group, and let a∈G. For n≥1,let G[n;a] := {x∈G:x^n =a}. Show that G[n; a] is either empty or equal to αG[n] := {αg : g ∈ G[n]}… [closed]

We were given questions to study for our exam coming up. We have not covered much of this topic, so any help would be greatly appreciated! Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, ...
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1answer
9 views

On ordered abelian groups containing $\mathbb{Z}$

Let $\Delta$ be an ordered abelian group containing $\mathbb{Z}$ as a subgroup of index $e$. I need to show that for any positive element $\delta \in \Delta$, we have $e\delta \geq 1$. I have no ...
3
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1answer
52 views

Commutativity of a Lie algebra $\Rightarrow$ the Lie group is abelian

Let $G$ be a Lie group, $\mathfrak{g}$ it's Lie algebra. Assume $[x,y]=0 \, \, \forall x,y \in \mathfrak{g}$. Is it true that $G$ is abelian? Remarks: (1) The other direction ($G$ abelian ...
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3answers
63 views

$H=\{x^2:x\in G\}$ then $H\unlhd G$

We have $H<G$ and element of $H$ is of the form $x^2$ where $x \in G$. H is a normal subgroup of $G$. The factor group $G/H$ is abelian. I tried first one by showing that $gh^2g^{-1} \in H$. ...
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26 views

Find the order of a subgroup of $\mathbb C^\times$

Let $\mathbb{C}^\times$ the multiplicative group of complex numbers different of zero. Let $H$ the subgroup $\mathbb{C}^\times$ of generated by $\{i, e^{\frac{2i \pi}{5}}, -1\}$. Find the order of ...
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1answer
45 views

Why is $\Bbb{Z_2} / \{e\} = \Bbb{Z_2}$?

Let the group $\Bbb{Z_2} = \{e, a\}$. We are given the quotient group $\Bbb{Z_2} / \{e\}$. So this gives us a set of left cosets: $\Bbb{Z_2} / \{e\} = \{e\{e\}, a\{e\}\} = \{\{e\}, \{a\}\} \neq \{e, ...
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0answers
55 views

Number of isomorphism classes of abelian groups of any order

Let $N$ be the order of an abelian group. The prime factorization is given by $N=\prod_{i=1}^{n}p_{i}^{e_{i}}$ with $p_{1}< p_{2}< \dots <p_{n}$ and $e_{i}\geq 1$. Let $\pi(n)$ denotes the ...
2
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4answers
316 views

Prove that a group is cyclic [closed]

$G$ is abelian of order $35$. and for all $x\in G$, $x^{35}=e$. I need to show that $G$ is cyclic. This seems perfectly obvious but I dont know how to write the proof. Help would be appreciated! ...
0
votes
2answers
97 views

How can you tell whether two groups are homomorphic/isomorphic? [closed]

Suppose you have two groups, $G$ and $H$. I've been taught the following definitions: "$G$ is homomorphic to $H$ iff there exists some function $\theta$ which gives the mapping $\theta : G ...