Should be used with the (group-theory) tag. A group $(G,*)$ is said to be abelian if $a*b=b*a$ for all $a,b\in G.$

learn more… | top users | synonyms

2
votes
1answer
19 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 ...
3
votes
1answer
243 views

normal p-subgroups of a finite group and chief factor

Let $G$ be a finite solvable group. Let $K/H$ be a chief factor of $G$ that is not of prime order, where $K$ is a $p$-subgroup of $G$ for some prime $p$ divides the order of $G$. Let $S$ be a proper ...
0
votes
1answer
35 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
vote
1answer
30 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 ...
1
vote
1answer
61 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.) ...
1
vote
0answers
35 views

Is this proof correct? concerning abelian groups

The aim of this question is to prove the existence of a bijection between two finitely generated Abelian groups $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$ and $D=(S^{r}, +)⊕(T,+)$ defined below: Proof: Let $C$ ...
2
votes
1answer
436 views

Subgroups and quotient groups of finite abelian $p$-groups

Motivation The fundamental theorem of finite abelian groups gives us a concise description of the isomorphism types of finite abelian $p$-groups $G$ (in the following, $p$ is a fixed prime). The ...
1
vote
0answers
27 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 ...
1
vote
0answers
28 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 ...
9
votes
4answers
373 views

Additive group of rationals has no minimal generating set

In a comment to Arturo Magidin's answer to this question, Jack Schmidt says that the additive group of the rationals has no minimal generating set. Why does $(\mathbb{Q},+)$ have no minimal ...
0
votes
1answer
32 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?
1
vote
1answer
86 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 ...
1
vote
1answer
83 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 $(\Bbb Z,+)$?
6
votes
1answer
70 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 ?
1
vote
1answer
41 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 ?
1
vote
0answers
33 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
votes
1answer
52 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
votes
4answers
199 views

Whether two quotients of $\mathbb{Z}^2$ are isomorphic.

Let $H_1$ be the subgroup of $\mathbb{Z}^2$ generated by $\{(1,2),(4,1)\}$, let $H_2$ be the subgroup of $\mathbb{Z}^2$ generated by $\{(3,2),(1,3)\}$. Is it true that $\mathbb{Z}^2/H_1\cong ...
3
votes
3answers
73 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
vote
1answer
25 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 ...
8
votes
3answers
379 views

Finding an explicit isomorphism between $\mathbb{Z}^{\times}_n$ and $\mathbb{Z}^{\times}_{2n}$

For an odd integer $n$, find an explicit isomorphism between $\mathbb{Z}^{\times}_n$ and $\mathbb{Z}^{\times}_{2n}$. How do I do this? I don't really know where to start. I can easily find bijections ...
8
votes
2answers
351 views

Does every ring with unity arise as an endomorphism ring?

I don't believe that every ring with a $1$ is the endomorphism ring of an abelian group but I currently don't see how to produce a counterexample.
3
votes
2answers
42 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 ...
2
votes
3answers
53 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 ...
1
vote
2answers
28 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 ...
1
vote
1answer
19 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
votes
1answer
53 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}\} = ...
1
vote
1answer
36 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 ...
0
votes
0answers
32 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 ...
8
votes
3answers
968 views

A nonsplit short exact sequence of abelian groups with $B \cong A \oplus C$

A homework problem asked to find a short exact sequence of abelian groups $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ such that $$B \cong A \oplus C$$ although the sequence does ...
4
votes
2answers
56 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
43 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
vote
1answer
23 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: ...
2
votes
2answers
92 views

On finite exponent abelian $p$-groups

Let $G$ be an abelian $p$-group non-isomorphic to any group of the form $H\times K$ where $H$ and $K$ are nontrivial groups. And let $\{|a|\mid a\in G\}$ have an upper bound in $\Bbb N$ . Is $G$ ...
-1
votes
3answers
61 views

$G$ is an abelian group of order a product of distinct primes $\implies G$ is cyclic?

If $G$ is an abelian group of order $p_1p_2...p_k$ , where $p_1,p_2,...,p_k$ are distinct primes , then is it true that $G$ is cyclic ?
6
votes
3answers
101 views

Automorphisms of $\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p$

Consider the abelian group $$G = \underbrace{\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p}_{n},$$ where $p$ is prime and $1\le n \le p$. I want to show that $G$ has no automorphism of order $p^2$. I ...
0
votes
3answers
582 views

What does it mean for a group to be Abelian?

I'm revising questions on groups for exams, and I still can't quite understand what an Abelian group is. Please help me understand, if anyone could give me a more simple explanation.
0
votes
2answers
46 views

Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
6
votes
1answer
83 views

About translating subsets of $\Bbb Z.$

This is a continuation of About translating subsets of R2. Is it possible to find a pair of sets $A,B\subseteq\Bbb Z$ such that A is a union of translated (only translations are allowed) copies of ...
1
vote
1answer
43 views

Are these exactly the abelian groups (2)?

This is a continuation of Are these exactly the abelian groups? I would like to consider another condition on a group and see if it implies commutativity. The condition is $$(\forall A,B\subseteq ...
0
votes
1answer
46 views

Under what conditions can the symmetric group be isomorphic to the abelian group?

The symmetric group is the set of all permutations. My question addresses the representability of the symmetric group using only additions. I am guessing that on the finite field $\mathbb{Z}/n ...
1
vote
2answers
29 views

Rank($U$) = Rank($U^2$) for group of units $U$

I am reading the paper "Algebraic Integers on the Unit Circle" by Ryan C. Daileda (http://www.sciencedirect.com/science/article/pii/S0022314X05002027). I am confused about how he concludes that the ...
2
votes
2answers
52 views

How to count the number of elements of given order?

I am trying to prove the following result. Let $G$ and $G'$ be two finite abelian groups. Besides, they have the same number of elements of any given order. Prove that $G\cong G'$. My attempt is ...
2
votes
4answers
174 views

there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$

there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$ But i don't know why it is true. should i investigate all group homomorphisms from $\mathbb Z\times\mathbb ...
7
votes
1answer
158 views

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic.

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic. You may only assume the Feit-Thompson theorem here and prove in the following ...
3
votes
3answers
165 views

Basic Group Theory question

This is not so much a plea of ignorance, but rather me trying to see whether intuitively I actually understand what is going on in group theory. The question asks What group is ...
0
votes
0answers
48 views

Finite generated abelian group $G$ and $H<G$. What is the rank of $(G/H)/(G/H)_t$?

I saw another question about this problem here. However there are quite different answers from my expectation. Anyway, here are my trials. Trial 1 : By structure theorem, $G\cong G_t\oplus F_1$ ...
0
votes
1answer
52 views

generators of groups from exact sequence

Suppose I have a middle term exact sequence of finitely generated abelian groups $G \longrightarrow H \longrightarrow K$. How do I get the generators of $H$ if I know the same for other two groups?
0
votes
3answers
73 views

A question on Abelian Groups

Prove that every subgroup of an Abelian group is Abelian but the converse is not true. I recently stumbled onto this question , but not able to solve it . Please help me out!
2
votes
1answer
52 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...