Tagged Questions

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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0
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3answers
21 views

Isomorphisms in finite abelian groups 1

True of false? If G and H are two groups with the same order and both are abelian, then they are isomorphic.
0
votes
1answer
36 views

All Isomorphic Classes of Abelian Groups of Order $n$

I know that each finite abelian group is isomorphic to a direct product of cyclic groups of prime orders $> 1$. This means taking a finite abelian group of order $n$, I can find the prime ...
1
vote
0answers
50 views

Verifying proof that set of all group homomorphisms is an abelian group

I'm working on a proof to show that for a group $G$ and an abelian group $H$, the set of all homomorphisms $\def\Hom{\operatorname{Hom}}\Hom(G,H)$ from $G$ to $H$ is an abelian group. I just want to ...
0
votes
0answers
21 views

question in finite abelian group [on hold]

Let $H$ be a subgroup of a finite abelian group $G$ . How to prove $G$ has a subgroup that is isomorphic to $\;H/G\;$ .(at W.Hungerford Algebra 2.2)
1
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0answers
50 views

Prove that module has finitely many elements

Let $p$ be a prime number. Consider the subring $U:= \mathbb{Z}[1/p]$ of $\mathbb{Q}$ and define the $\mathbb{Z}$-module $M:=U/ \mathbb{Z}$ (1): Show that any $\mathbb{Z}$-submodule of $M$ that is ...
1
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0answers
10 views

Orthogonality Relations for Character Groups

I'm trying to understand a part of a proof of orthogonality relations for character groups and finite abelian groups, and I don't quite get this part from the below link: ...
-6
votes
0answers
24 views

a question about finite abelian group [on hold]

Let $H$ be a subgroup of a finite abelian group $G$ . Show that $G$ has a subgroup that is isomorphic to $H/G$.
2
votes
3answers
119 views

Identity and Inverse Homomorphisms

For a group G and an abelian group H, Hom(G,H) is the set of all homomorphisms from G to H. My notes from class talk about the identity and the inverse homomorphism- I was wondering what these are? ...
0
votes
0answers
11 views

Prove the subgroup <h.v> generated by h and v is Abelian and of order 9

I'm working on flushing out Theorems, Lemmas, Definitions, etc for a paper on the Rubik's Slide. (So I'll probably have more questions to come.) $\underline {What\ I\ Need\ To\ Know:}$ How to show ...
2
votes
1answer
43 views

Can we define a binary operation on $\mathbb Z$ to make it a vector space over $\mathbb Q$?

It is known that any infinite cyclic group , in particular $(\mathbb Z, +)$ , can never be a vector space . So we may ask , Can we define an operation $*$ on $\mathbb Z$ such that $(\mathbb Z , *)$ ...
0
votes
1answer
23 views

Number of mutually non isomorphic Abelian groups

Let p and q be distinct primes. How many mutually non-isomorphic Abelian groups are there of order p^2q^4. I think there are 6 of them: p^2q^4 q, qp, q^2p q^2, q^2p^2 p, pq^3 pq, pq^3 q, q^3p^2 in ...
2
votes
2answers
48 views

Epimorphism that is not surjective in the category of Torsion Free Abelian Groups

In reading about cokernels (relating to a homework question I have) I came across the following: https://www.dpmms.cam.ac.uk/~jg352/pdf/CTSheet4-2013.pdf I specifically wondered about question 5a. ...
1
vote
1answer
49 views

Free group on two generators and commutators. Why it's enough to add the relation ab=ba?

I've looked through lots of question on this topics, but I cannot find what I want to prove: I've seen in a lots of exercises sheets that the abelianization of a free group with two generators (let's ...
3
votes
0answers
72 views

A group is abelian [duplicate]

Let $G$ be a finite group. For any two elements $a,b \neq e\in G$ there exits an automorphism $\sigma$ such that $\sigma(a)=b$. Prove that $G$ is abelian. Only thing that I could conclude about the ...
0
votes
1answer
28 views

Finite Abelian Group Proof

Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ for some prime $p$. I'm not sure what to do. Any proofs or hints ...
3
votes
3answers
107 views

Is every free subgroup a direct summand?

Let $G$ be an abelian group. Suppose that $F$ is a subgroup of $G$ such that $F$ is free. Does there necessarily exist a subgroup $H\subset G$ such that $G\cong F\oplus H$? Motivation: In Lang's ...
2
votes
1answer
61 views

homomorphisms of abelian groups

Describe: 1) Hom(Q/Z->Q) 2) Hom(Q->Q/Z) Q rational numbers Z integers My thoughts 1) Q/Z i can describe as a/b, a,b coprime and smaller than 1. So I thought Hom(Q/Z->Q) I can describe as [f:a+N->b, N ...
0
votes
4answers
29 views

Automorphism iff G is abelian

Let $G$ be a group. Prove the mapping $\alpha(g)=g^{-1}\forall g \in G$ is an automorphism iff $G$ is abelian. Proof (forwards): Assume $G$ is an automorphism. Show $ab=ba$. How would I even go about ...
2
votes
3answers
82 views

How many distinct subgroups of order 10 are there in a non-cyclic abelian group of order 20?

We are currently working with free abelian groups and finitely generated groups. The homework problem asks us to find the number of distinct subgroups of order 10 in a non-cyclic abelian group of ...
1
vote
0answers
32 views

Abelian Group (Alternative Proof)

Is there an alternative method to prove $(ab)^{2}=a^{2}b^{2}$ for all elements $a,b \in G \implies$ $(ab)^{-1}=a^{-1}b^{-1}$ for all elements $a,b \in G$. then the one I give below? Let $G$ be ...
1
vote
2answers
95 views

Order of the group.

An abelian group $G$ is generated by $x$ and $y$ with $$O(x)=16,O(y)=24,x^2=y^3$$ What is the order of $G$? My attempt:There are $24+16-1=39$ elements generated by $x$ and $y$ separately. Also ...
1
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0answers
70 views

Description of an abelian group

I'm again stuck in an algebra exercise. I'm not sure if I understand the problem right. Could it be that I have to show that $\mathbb{Z}[i]/\gamma$ can be expressed by a product of finite abelian ...
1
vote
1answer
30 views

Can we calculate the order of $\hom (G,G')$ in terms of $|G| $ and $ |G'|$ , when $G,G'$ are finite abelian groups?

Let $G,G'$ be abelian groups and let $\hom (G,G')$ be the set of all homomorphisms from $G$ to $G'$. We define an operation $\ast$ on $\hom (G,G')$ as: for $f,g \in \hom(G,G') \space , (f\ast ...
0
votes
2answers
34 views

|ab|=lcm(|a|,|b|) in an abelian group

Assume in an abelian group $G$ that $\langle b\rangle\cap \langle a\rangle=e$, then the order of $(ab)$ is the lcm of the orders of $a$ and $b$. Essentially, $|ab|=\operatorname{lcm}(|a|,|b|)$. So ...
3
votes
2answers
74 views

Irreducible subgroups of the additive rationals

Let $G$ be a group. A proper subgroup $H$ is called irreducible if $H$ can't be written as an intersection of two subgroups which contain it properly. I'd like to know if $(\mathbb Q,+)$ (and ...
1
vote
2answers
37 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
votes
1answer
78 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 ...
4
votes
1answer
140 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
vote
1answer
26 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
votes
1answer
23 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 $n=1$.
-2
votes
2answers
60 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.
1
vote
0answers
28 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
votes
1answer
153 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
votes
1answer
37 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
vote
2answers
21 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
votes
1answer
38 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 ...
1
vote
3answers
57 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
votes
0answers
48 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
votes
1answer
28 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 ...
0
votes
0answers
40 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 ...
1
vote
0answers
29 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
votes
1answer
27 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
votes
4answers
155 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
votes
2answers
37 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
2answers
60 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
votes
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 ...
1
vote
6answers
421 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
votes
1answer
39 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
votes
1answer
34 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
votes
1answer
30 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 ...