Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$
5
votes
3answers
268 views
Prove that if $g^2=e$ for all g in G then G is Abelian.
This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem to crack it.
(please note that $e$ in the question is the ...
16
votes
4answers
2k views
Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian
I've been working on this problem listed in Herstein's Topics in Algebra (Chapter 2.3, problem 4):
If $G$ is a group such that $(ab)^i = a^ib^i$ for three consecutive integers $i$ for all $a, b\in ...
31
votes
3answers
1k views
The direct sum $\oplus$ versus the cartesian product $\times$
In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines $A^n := A\times A\times\ldots\times ...
19
votes
1answer
434 views
Recovering a finite group's structure from the order of its elements.
Suppose you know the following two things about a group $G$ with $n$ elements:
the order of each of the $n$ elements in $G$;
$G$ is uniquely determined by the orders in (1).
Question: How ...
3
votes
1answer
1k views
If $G/Z(G)$ is cyclic, then $G$ is abelian
Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer):
Prove that if $G/Z(G)$ is ...
18
votes
2answers
960 views
Structure Theorem for abelian torsion groups that are not finitely generated
I know about the structure theorem for finitely generated abelian groups.
I'm wondering whether there exists a similar structure theorem for abelian groups that are not finitely generated. In ...
12
votes
2answers
132 views
Status of the classification of non-finitely generated abelian groups.
From the Wikipedia on abelian groups:
By contrast, classification of general infinitely-generated abelian groups is far from complete.
How far are we from a classification exactly? It seems ...
7
votes
3answers
376 views
Finite abelian $p$-group with only one subgroup size $p$ is cyclic
My goal is to prove this:
If $G$ is a finite abelian $p$-group with a unique subgroup of size $p$,
then $G$ is cyclic.
I tried to prove this by induction on $n$, where $|G| = p^n$ but was not ...
6
votes
2answers
287 views
Whence this generalization of linear (in)dependence?
I recently came across a definition of (in)dependence that is supposed to be a generalization of linear (in)dependence among a set of vectors:
An element $x$ is dependent on a set of elements ...
4
votes
2answers
419 views
A condition for a subgroup of a finitely generated free abelian group to have finite index
Let A be a free Abelian group of finite rank and B be a subgroup of A such that $A=B+pA$ for some prime number p, then how to prove $B$ is a subgroup of finite index in A? And if $A=B+pA$ holds for ...
-3
votes
2answers
371 views
Non-isomorphic abelian groups of order $19^5$
I am trying to classify abelian groups of order $19^5$ up to isomorphism. Can anyone provide any approaches or hints?
7
votes
1answer
148 views
Isomorphism Types
I'm a little panicking right now. I have finals soon, and I don't know how to go about solving this: Classify the isomorphism types of abelian groups of order 44. Solutions or even hints would be ...
2
votes
4answers
303 views
Why is the set of natural numbers considered a non-Abelian group?
I don't understand why the set of natural numbers constitutes a commutative monoid with addition, but is not considered an Abelian group.
1
vote
0answers
186 views
Normal abelian subgroup of a solvable group [duplicate]
Possible Duplicate:
A Nontrivial Subgroup of a Solvable Group
How to find a normal abelian subgroup in a solvable group?
Could someone help me with this proof? Let $G$ be a solvable group ...
14
votes
4answers
677 views
How many non-isomorphic abelian groups of order $\kappa$ are there for $\kappa$ infinite?
Let $\kappa$ be an infinite cardinal. How many non-isomorphic abelian groups of order (cardinality) $\kappa$ are there?
For finite $\kappa,$ we can use the classification theorem and obtain the ...
8
votes
5answers
944 views
Are cyclic groups always abelian?
If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is?
Thanks in advance!
9
votes
1answer
141 views
Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?
Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?
I know that there is a bijection between $\mathbb{R}$ and $\mathbb{C}$, and this question asks whether they are isomorphic as ...
1
vote
2answers
223 views
Factor groups and Torsion subgroups
If $G$ is abelian, $T(G)$ is the torsion subgroup, then $G/T(G)$ is torsion free.
If $T(G) = \{1\}$, then $G$ is called a torsion-free group. Below is what I did to prove this statement.
...
8
votes
3answers
131 views
Tensor-commutative abelian groups
Say that an abelian group $A$ is tensor-commutative if the equality $x\otimes y=y\otimes x$ holds in $A\otimes_{\mathbb Z}A$ for all $x,y$ in $A$.
The first question is somewhat vague:
Question 1. ...
7
votes
3answers
185 views
The Hopfian property for groups
Let $G$ be a group, which for my purposes would be abelian. To say that $G$ has the Hopf property is to say that every epimorphism of $G$ is an automorphism. Does anyone happen to recall the context ...
5
votes
1answer
121 views
For which $n$, $G$ is abelian?
My question is:
For Which natural numbers $n$, a finite group $G$ of order $n$ is an abelian group?
Obviouslyو for $n≤4$ and when $n$ is a prime number, we have $G$ is abelian. Can we consider ...
4
votes
2answers
189 views
Elements of finite order in the group of arithmetic functions under Dirichlet convolution.
Let $(G, ∗)$ be the group of arithmetic functions $f : N \to C$ that satisfy $f (1)\neq 0$, with group operation given by the Dirichlet product $∗$. The identity function $I$ is the identity element ...
3
votes
2answers
318 views
Structure theorem for finitely generated abelian groups
How can we use fundamental theorem of finitely generated abelian groups to list all abelian groups of order 16 up to isomorphism.
10
votes
1answer
351 views
Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?
I think the answer is yes.
Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then ...
8
votes
1answer
305 views
Does the splitting lemma hold without the axiom of choice?
In part of the proof of the splitting lemma (a left-split short exact sequence of abelian groups is right-split) it seems necessary to invoke the axiom of choice. That is, if $0\to A\overset{f}{\to} ...
6
votes
2answers
90 views
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
Assume that $\pi$ is an isomorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|>\frac{3|G|}{4}$, then $G$ is an abelian group. Anyone ...
5
votes
2answers
364 views
A Counter example for direct summand
All my attempts proving the following claim have been useless and it seems to be wrong, but can not find any counter example(s) for it. :)
"If U and N be two direct summands of an abelian group G ...
3
votes
1answer
69 views
Inductive vs projective limit of sequence of split surjections
Let
$$
A_1\twoheadrightarrow
A_2\twoheadrightarrow
A_3\twoheadrightarrow
A_4\twoheadrightarrow
\cdots
$$
be an inductive sequence of abelian groups, the connecting homomorphisms of which are ...
3
votes
1answer
125 views
When abelian group is divisible?
By definition, group $G$ is divisible if for any $g\in G$ and natural number $n$ there is $h\in G$ such that $g=h^n$.
Let $A$ be abelian group with no proper subgroups of finite index. How can I prove ...
3
votes
3answers
184 views
Check if $(\mathbb Z_7, \odot)$ is an abelian group, issue in finding inverse element
Take $\mathbb Z_7$ and the operation $\odot$ defined on it as follows $\forall a,b \in \mathbb Z_7$:
$$\begin{aligned} a \odot b=a+b+3\end{aligned}$$
Check if $(\mathbb Z_7, \odot)$ is a group and ...
3
votes
1answer
214 views
how to show that a group is elementarily equivalent to the additive group of integers
Is there any fairly easy way of showing a group is elementarily equivalent to the additive group of the integers?
I've found a simple characterization here: A ‘natural’ theory without a prime model, ...
2
votes
1answer
78 views
Verifing some properties about $G$
I have the following problem$^*$:
Prove that the group $G$ having generators and relations respectively $$X=\{x_0,x_1,x_2,\ldots\} \\\{px_0=0,x_0=p^nx_n, \text{all } n\geq1\}$$ is an infinite ...
2
votes
1answer
709 views
Computing the Smith Normal Form
This question is related to the Smith Normal Form of Matrices:
Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix
$R :=$
$$
\begin{bmatrix}
-6 & 111 & ...
1
vote
0answers
71 views
The following groups are the same.
This is an exercise from J.J.Rotman's book:
Prove that the following groups are all isomorphic:
$$G_1=\frac{\mathbb R}{\mathbb Z},G_2=\prod_p{\mathbb Z(p^{\infty})}, G_3=\mathbb ...
1
vote
1answer
68 views
Divisible abelian $q$-group of finite rank
What does "finite rank" mean in the context of divisible abelian $q$-group?
A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer ...
1
vote
5answers
214 views
How to prove a group has a basis with exactly one element?
I am struggling with the following question.
Suppose I have a group $H$ which is a subgroup of $\mathbb{Z}\oplus\mathbb{Z}$, such that any element $\begin{bmatrix}
a \\[0.3em]
b ...
1
vote
1answer
406 views
Subgroups of a finitely generated abelian group without torsion
If $G\cong \mathbb{Z}\times \mathbb{Z}\times \dots \times \mathbb{Z}$ is a finitely generated abelian group without torsion of rank $n$, where $n$ is the number of copies of $\mathbb{Z}$.
Then any ...
1
vote
1answer
108 views
Computing Quotient Groups
Let $G/H = \mathbb{Z}_{4} \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$. I know that $|G/H|$ = 4, so $G/H \simeq \mathbb{Z}_{2} \times \mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$. Since $G/H$ has an ...
1
vote
5answers
271 views
A criterion for a group to be abelian
I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all.
Let and ...
7
votes
1answer
77 views
Finding the order of the automorphism group of the abelian group of order 8.
So I am given an abelian group of order $8$ such that for all non-identity elements $x^2 = e$ (all elements have order two). So I know the answer is gonna be $168$, but I gotta prove this.
So far I ...
6
votes
2answers
590 views
If $H$ is a normal subgroup of $G$ and if both $H$ and $G/H$ are abelian, is $G$ abelian?
Pretty straightforward:
If $H$ is a normal subgroup of $G$ and if both $H$ and $G/H$ are abelian, is $G$ abelian?
5
votes
1answer
178 views
If L is a subgroup of $\mathbb{Z}^{3}$, linearly independent, linear equations
This exercise is from a book called "Introduction a L'Algebre et L'Analyse Modernes" de M.Zamansky, I attempted to solve. But I don't know if my solutions are correct (they seem too short to be ...
4
votes
1answer
226 views
Non-trivial homomorphism between multiplicative group of rationals and integers
Let $\mathbb{Q}^{\times}$ be the multiplicative group of non-zero rationals. Is there a non-trivial homomorphism $\mathbb{Q}^{\times} \to \mathbb{Z}$? In the same spirit, is there a homomorphism ...
4
votes
1answer
125 views
Smallest pure subgroup containing a fixed subgroup
I will ask a slightly more precise question then in the title.
Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups they generate are in direct sum $\langle g_1 ...
3
votes
1answer
72 views
The free abelian group monad
The forgetful functor $U : \mathsf{Ab} \to \mathsf{Set}$ is monadic, this follows from Beck's monadicity theorem or some other general result. Anyway, I would like to prove this directly, thereby ...
3
votes
1answer
148 views
An indecomposable $\mathbf{Z}$-module whose injective hull is not indecomposable
I'd like to find an indecomposable $\mathbf{Z}$-module whose injective hull is not indecomposable, and I'm running out of ideas:
The only indecomposable $\mathbf{Z}$-modules I know are $\mathbf{Z}$, ...
3
votes
1answer
154 views
Finding subgroups of index 2
Let $G = \prod_{i=1}^\infty \mathbb{Z}_2$ with addition mod 2. I am trying to find subgroups of index 2. I see that taking the entire space and removing all sequences which have a 1 in a certain ...
3
votes
1answer
287 views
Computing index of a subgroup of a free abelian group
We looked briefly at this example in class but I'm not quite sure how to proceed, and I can't find examples of this in any textbooks I have (Dummit & Foote and Nicholson).
Suppose we have $H = ...
3
votes
4answers
178 views
$F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$
I am confused by the proof a proposition:
$F$ is a free abelian group on a set $X$ and $H$ is a subgroup of $F$, then $H$ is free abelian on a set $Y$, where $|Y| \leq |X|.$
The proof is:
...
2
votes
0answers
45 views
Inductive vs projective limit of sequence of split surjections II
This question is a follow-up of this earlier question I asked.
Let
$$
A_1\twoheadrightarrow
A_2\twoheadrightarrow
A_3\twoheadrightarrow
A_4\twoheadrightarrow
\cdots
$$
be an inductive sequence of ...
