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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2answers
45 views

Least common multiple of orders and abelian groups.

I am a little stuck here and would like some minor help. The quesiton I am dealing with is: Assume in an abelian group G that $<b>{\large\cap} <a>=e$, then the order of $(ab)$ is the lcm ...
0
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1answer
92 views

$G$ infinite abelian group with $[G:H]$ finite for every non trivial subgroup $H$ , to prove $G$ is cyclic

Let $G$ be an infinite abelian group such that for any non-trivial subgroup $H$ of $G$ , $[G:H ]$ is finite ; then how to prove that $G$ is cyclic ? Please don't use any structure theorem of abelian ...
5
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1answer
232 views

Direct sum and direct product of infinitely many abelian groups are not isomorphic

Let $I$ be an infinite set, and for each $i$ let $A_i$ be an abelian group with order $o(A_i) \ge 2$. Prove that the direct product $\prod A_i$ and the direct sum (coproduct) $\bigoplus A_i$ are ...
1
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1answer
27 views

Show $a\in G$ is contained in $Z(G)$ iff $Z(a)=G$ for center and centralizer? [closed]

The center of a group $G$ is defined as the set $Z(G):= \{a\in G\mid \forall b\in G : ab=ba\}$ and the centralizer of an element $a\in G$ is defined as the set $Z(a) := \{b\in G\mid ab=ba\}$. How can ...
0
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1answer
24 views

The Dihedral group $D_1$ is non-abelian?

Same as above. I'm trying to show that for any n being odd, $D_n$ has exactly n elements of order 2 where $D_n$ is non-abelian. I know that for $n\ge3$ this is true, but what about for $n=1$.
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2answers
73 views

Two abelian groups with the same order are isomorphic? [closed]

True of false: if G and H are two groups with the same order and both are abelian, then they are isomorphic.
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0answers
32 views

Measure theory mapping sets to groups?

This is a question from a physicist wondering if a certain idea in mathematics has been developed. Intuitively, suppose I have a number of objects distributed in space. I want a function that given a ...
-1
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1answer
167 views

On groups with none of their quotient groups divisible [closed]

Does there exist a group $G$ that satisfies the following conditions: Any proper subgroup of $G$ is contained in a maximal subgroup. There is some $N\unlhd G$ such that $\frac{G}{N}$ is divisible. ...
0
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1answer
40 views

Rank of abelian groups

I have read that given a $\mathbb{Z}$-module $M$, the maximal number of $\mathbb{Z}$-linear independent elements is given by $\operatorname{rank}M=\dim_\mathbb{Q}(\mathbb{Q}\otimes_\mathbb{Z}M)$. ...
1
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2answers
23 views

Clarification on finding another subgroup given the order of two existing subgroups

If we assume that G is abelian and that it has a subgroup of order 7 and another of order 11. If we were asked to find another subgroup of this group would we take the least common multiple of the ...
0
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1answer
43 views

Subgroup of an abelian group isomorphic to a given quotient group

STATEMENT: Let $H$ be a subgroup of a finite abelian group $G$. Show that $G$ has a subgroups that is isomorphic to $G/H$. QUESTION: Could someone offer a proof using dual groups. I have found one ...
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3answers
85 views

Prove that no finite abelian group is divisible.

A nontrivial abelian group $G$ is called divisible if for each $a \in G$ and each nonzero integer $k$ there exists an element $x \in G$ such that $x^k=a$. Prove that no finite abelian group is ...
0
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0answers
63 views

Number of Abelian Groups of Order 36

GRE Subject Test Question: Up to isomorphism, how many abelian groups are there of order 36? The answer given is 4 and the explanation is as follows: Let G be an abelian group with order n. Then G ...
0
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1answer
32 views

Submodule iff subgroup?

It is late at night and time for another silly question: Is it true that a subset $S$ of an $R$-module $M$ is a submodule if and only if it is a subgroup of $M$ as an abelian group? Of course, by ...
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0answers
45 views

Finding an isomorphism from $\mathbb{R}^\times$ to a defined group $G$

Here's the problem I am solving: $G=\{x\in \mathbb{R}:x\not = 0\}$. The operation for $G$ is "$*$", with $x*y=\frac{1}{2}xy.\mathbb{R}^\times$ is the multiplicative group $\mathbb{R}.$ Find an ...
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0answers
34 views

Proof of elliptic curves being an abelian group

What are some simple proofs that the points on an elliptic curve form an abelian group under addition? I am mostly looking for proofs of closure and associativity, since the other three requirements ...
0
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1answer
31 views

Proving that some property on a chain complex of groups implies isomorphism between direct sums of these groups.

Let $C_*$ be a chain complex such that every $C_i$ is a torsion-free finitely generated abelian group, with $C_i=0$ for every $i<0$ and every $i>N$ for some sufficiently large integer $N$. If ...
0
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4answers
198 views

Show that an abelian group $G$ of order 55 must be cyclic.

I know that in order to be cyclic: A group G is called cyclic if there exists an element g in G such that G = ⟨g⟩ = { $g^n$ | n is an integer } by wikipedia. But I just get lost in how simple it looks ...
0
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2answers
40 views

abelian groups?

let p and q be distinct prime numbers. how does the number (up to isomorphism) of abelian groups of order p^r compare with the number (up to isomorphism) of abelian groups of order q^r? I am just not ...
0
votes
3answers
69 views

Can $\mathbb{Z}/n\mathbb{Z}$ (not $(\mathbb{Z}/n\mathbb{Z})^{\times}$) be a group under multiplication?

I was wondering why we usually say $\mathbb{Z}/n\mathbb{Z}$ is a group under addition and invent notation like $(\mathbb{Z}/n\mathbb{Z})^\times$ specifically for the multiplicative group modulo $n$. ...
0
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2answers
46 views

Suppose G is a group which has only one element a such that |a| = 2. Prove that xa = ax, for all x ∈ G.

I know the following are true. 1) There is an inverse of a 2) There is an identity element (e*a) = a In this case, e = 1 and the inverse of a is 1/|2|. However, if a is the only element in G and a ...
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6answers
445 views

Let G be a group and a; b ∈ G. Suppose |a| = |b| = |ab| = 2. Then show that ab = ba.

I'm having trouble understanding this question and help would be appreciated. If |ab|=2 and |a|=2, |b|=2, wouldn't this imply that |a||b|=|ab|=4? How would I go about proving that this is Abelian? ...
0
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1answer
45 views

Order of Group with Elements of Order 2 [duplicate]

Let G be a finite group such that every element in G which isn't the identity has order of 2. Show that $|G| = 2^{n}$ for some $n \in \mathbb{N}$. I know that G is necessarily going to be abelian. ...
0
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1answer
35 views

The number, up to isomorphism, or abelian grips of order 40 is

The number, up to isomorphism, or abelian groups of order 40 is: I got: 2*2*10 2*20 40 So the total number is 3. However, the answer says 7, where 40 10*4 8*5 20*2 10*2*2 5*4*2 I think the ...
2
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1answer
31 views

Property of abelian groups without using Lagrange's theorem

I need to prove the following without using Lagrange's Theorem: Show that for an abelian group $G$, $\forall \; a \in G:$ $a^{o(G)}=e$ . This is a generalization of the Euler-Phi Theorem. So I ...
0
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1answer
41 views

Groups of Order 2 with subgroups

Let G be an abelian group and $a,b\in G$ be two distinct elements with a and b or order $2$. Show that $H=\{e,a,b,ab\}$ forms a subgroup and write out its multiplication table. Justify why all the ...
1
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1answer
43 views

$(\mathbb C^{\times}, \cdot)$ is a subgroup of $(GL(n,\mathbb C), \cdot)$

I am learning groups and subgroups in my algebra course. Today, we talked about examples of subgroup but I am not sure why the following holds: $(\mathbb C^{\times}, \cdot)$ is a subgroup of ...
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0answers
32 views

Direct limits of free abelian groups and diagonalization

So, say I have a matrix $A\in M_d(\mathbb{Z})$ and would like to describe the group $\lim(\mathbb{Z}^d,A)$, i.e. the limit of the stationary system $$ \mathbb{Z}^d\to^A \mathbb{Z}^d \to^A ...
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0answers
29 views

$G$ a finite group $n$-abelian goup and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then to show $G$ is abelian [duplicate]

Let $G$ be a finite group and $n$ be a given positive integer such that $(ab)^n=a^nb^n , \forall a,b \in G$ and g.c.d.$\big(|G|,n(n-1)\big)=1$ , then how to prove that $G$ is abelian ? If I can show ...
1
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1answer
80 views

Complement but not direct summand; Lam, Lectures on Modules and Rings, Example 6.17(5)

Let $S$ be a submodule of an $R$-module $M$. A submodule $C⊆M$ is said to be a complement to $S$ (in $M$) if $C$ is maximal with respect to the property that $C∩S=0$. (This does exist by Zorn Lemma.) ...
0
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1answer
40 views

$R$ is not a direct Sum of its subgroups

How to prove the set of real numbers under addition i.e $(R,+)$ is not the direct sum of two of its proper subgroups?
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1answer
89 views

When an Abelian group is cyclic

Let G be a finite abelian group.It contains a non trivial subgroup which is contained in every non trivial subgroup.Then G must be cyclic. This is a problem of Herstein book(Pg 108,#11 2nd edition).I ...
6
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1answer
78 views

$G$ is torsion-free $[G:Z(G)]$ is finite $\implies$ $G$ is abelian ?

If $G$ is a a group having no non-identity element of finite order and $Z(G)$ , the center of the group , has finite index , then is it true that $G$ is abelian ?
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1answer
45 views

From Automorphism to abelian ness … in a finite group

Let $G$ be a finite group such that for any two non-identity elements $a,b$ in $G$ , there is an Automorphism of $G$ sending $a$ to $b$ , then is it true that $G$ is abelian ?
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0answers
39 views

divisible subgroup without axiom of choice

the theorem asserting that the divisible subgroup of an Abelian group is a direct summand depends on Zorn's lemma. in ZF without AC is there a construction which yields a model of an Abelian group ...
2
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1answer
68 views

Exercise on characterization of free abelian groups

I was wondering if someone can please check my work on a homework problem. This is from the graduate Hungerford text. Chapter 2.1, number 3. Let $X=\{a_i\ |\ i\in I\}$ be a set. Then the free abelian ...
3
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3answers
85 views

An epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$

I want an explicit example of an epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$. Thanks.
1
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1answer
55 views

Rational group algebras and maximal orders

Let $G$ be a finite group, and $\mathbb{Q}[G]$ be the rational group algebra. Then the group ring $\mathbb{Z}[G]$ is an order in $\mathbb{Q}[G]$, but is not in general a maximal order. What can we ...
3
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2answers
46 views

Abelian group and morphism equivalent statement

Exercise Show that the following statements are equivalent: $(i) \space G \space \text{is abelian.}$ $(ii) \space \text{the map f: G} \to \text{G defined as} \space f(x)=x^{-1} \space \text{is a ...
3
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3answers
69 views

When is a non-trivial homomorphism injective?

I noticed that over the natural numers $(\mathbb{Z},+)$ any group homomorphism $f : \mathbb{Z} \rightarrow \mathbb{Z}$ that is not the trivial one, is automatically injective. Where exactly does ...
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2answers
30 views

Does the following binary operation form a group on a set with 3 elements? (multiple identities?)

Let S = {a, b, c}. *| a b c ----------- a| a b c b| b a a c| c a a This seems to have all the desired characteristics of a group, however, both b and c ...
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1answer
111 views

A sequence of subsets of $\Bbb Z$ not containing nontrivial subgroups [closed]

Is there a sequence $(A_n)$ of subsets of $\Bbb Z$ such that always $\{a-b\mid a,b\in A_{n+1}\}$ is a proper subset of $A_n$ and no $A_n$ contains an infinite subgroup of $(\Bbb Z,+)$? (Ed.: this ...
1
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1answer
20 views

Tensor product of $\mathbb{R}^d$ and $\mathbb{R}^s$ as abelian groups

It is well known (and easy to prove) that $\mathbb{R}^d\otimes_{\mathbb{R}} \mathbb{R}^s$ is isomorphic as a vector space to $\mathbb{R}^{sd}$. Now, I would like to know a simple description of the ...
2
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1answer
58 views

About a nested sequence of subsets of integers

Let $(H_n)$ be a sequence of nonempty subsets of $\Bbb Z$ such that always $\{a-b\mid a,b\in H_{n+1}\}\subsetneqq H_n$. Can we deduce that there is some $n$ such that $\{a-b\mid a,b\in H_{n}\} = ...
0
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1answer
93 views

A sequence of subsets of $\Bbb Z$ [closed]

Is there a sequence $(A_n)$ of nonempty subsets of $\Bbb Z$ such that for each $n$, $$\{a-b\mid a,b\in A_{n+1}\}\subsetneqq A_n$$ ?
0
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0answers
33 views

If the automorphism group of a group is cyclic, then the group is commutative [duplicate]

Let $G$ be a group and the $Aut(G)$ group is cyclic $\Rightarrow$ the group $G$ is commutative. I looked at the homomorphism $\varphi : G \rightarrow Aut(G) \ g \mapsto (x \mapsto gxg^{-1})$. Let ...
1
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1answer
56 views

Torsion subgroup of $\mathbb{C}^\times$

I need to find the torsion subgroup of the multiplicative abelian group $\mathbb{C}^\times$. This is from a homework assignment sheet, and I'm not sure what the notation $\mathbb{C}^\times$ stands ...
4
votes
2answers
67 views

Is $<\mathbb Q^+, \times>$ the free abelian group on countably infinitely many generators?

It seems to make sense to me that it should be, with the generators being the set of primes. However, I'm not sure that my intuition is right. Additionally, would this not be contradicted by the fact ...
0
votes
1answer
50 views

easy short exact sequence question

Suppose I have have a short exact sequence of finitely generated Abelian groups $0 \longrightarrow G \overset{f}\longrightarrow H \overset{g}\longrightarrow K \longrightarrow 0$. Suppose I have a ...
1
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1answer
25 views

Show that it is torsion

I am trying to solve the following exercise Let $G$ be an abelian group, and let $S\subset G$ be a subgroup. If $H$ is maximal with $H\cap S=\{0\}$, prove that $G/(H+S)$ is torsion. My attempt: ...