Questions tagged [abelian-groups]

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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On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case. Let $M$ be a free abelian group and $N$ a ...
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90 views

Does $A$ have any special status as an $\mathrm{End}(A)$ module?

Let $A$ be an abelian group and $\mathrm{End}(A)$ its endomorphism ring. Then to give an abelian group $B$ the structure of a (left) $\mathrm{End}(A)$ module, we provide a morphism $\mathrm{End}(A)\to\...
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205 views

Solving particular type of system of equations in $\mathbb R/\mathbb Z$

I apologize in advance for the long post. You can freely skip to the last paragraph. I was motivated by this question given in 5th grade mathematics competition that I was solving with my advanced ...
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120 views

Abelian groups whose automorphism group is a $p$ group

$\def\Aut{\operatorname{Aut}}$ Let $G$ be a finite abelian group such that $\Aut(G)$ is an $p$ group ,that is, $|\Aut(G)|=p^n$ . Then can we determine the cyclic decomposition of $G$ or at least the ...
6
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70 views

Shift operator on locally compact groups

Assume $f:G\rightarrow H$ is a measurable function between two locally compact abelian groups and let $T^h(f) = f\circ T^h$, where $T^h(x) = x-h$ (group operations in G and H are written additively). ...
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66 views

Can we prove that $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$ is a cyclic group by using $p$-adic integer?

It is well known that $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$ is a cyclic group for a prime $p>2$ and $n\geq 1$. However, most of the proofs are a little complicated, and I want to find some neat ...
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60 views

The order of elements in groups of a particular structure

$\textbf{Assumptions:}$ Let $G$ be finite abelian group and $A = \{a_i\}_{i=1}^m \subseteq G.$ Suppose that every $g \in G$ can be formed uniquely using elements of $A$ as follows: $$g = \lambda_1 ...
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82 views

When is $\binom{m+s}{s}/(s+1)$ a natural number for $m, s \in \mathbb{N}$ and $m,s > 1?$

Background: I am currently working on a research project on the existence of certain spanning sets for finite abelian groups, $G$. In particular, I am looking at the existence of perfect $s$-bases of ...
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2k views

Using Lagrange's theorem, prove that a non-abelian group of order $10$ must have a subgroup of order $5$.

Using Lagrange's theorem, prove that a non-abelian group of order $10$ must have a subgroup of order $5$. Attempt: Let $G$ be a group of order $10$. By Lagrange's theorem, if there exist a subgroup $...
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144 views

Schröder-Bernstein for abelian groups with direct summands

What is a simple example of two abelian groups $A,B$ which are isomorphic to direct summands of each other (that is, $A \cong B + C$ and $B \cong A + D$ for some abelian groups $C,D$), but which are ...
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123 views

Generating Sets for Subgroups of $(\Bbb Z^n,+)$.

The question Finite Generated Abelian Torsion Free Group is a Free Abelian Group led me to conjecture and prove an interesting thing about generating sets for $\Bbb Z^n$ and certain subgroups. If $x=(...
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78 views

Graphing elliptical curves based on group operation

I just found this and it blew my mind (he gives an elliptical curve to do multiplication). If I understand correctly (from reading the link and other things) the Abelian group he is using is $\mathbb{...
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178 views

When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
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150 views

Possible subgroups of $\mathbb{Z}/3^6\mathbb{Z} \oplus\mathbb{Z}/3^5\mathbb{Z}\oplus\mathbb{Z}/3^2\mathbb{Z}$

$G \cong \mathbb{Z}/3^6\mathbb{Z} \oplus\mathbb{Z}/3^5\mathbb{Z}\oplus\mathbb{Z}/3^2\mathbb{Z}$ $H\leq G$ so that $G/H \cong \mathbb{Z}/3^2\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z} $ Find all possible $...
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76 views

Existence of abelian group which has no “square-root” but whose “cube” has a “square-root”

Does there exist an abelian group $G$ such that $G \ncong H \times H$ for every abelian group $H$ but $G \times G \times G \cong K \times K$ for some abelian group $K$ ? Also see Existence of ...
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47 views

What is the easiest way to find the number of isomorphic subgroup?

Find the number of subgroups of $\mathbb{Z}_{100}\times\mathbb{Z}_{500}$ are isomorphic with $\mathbb{Z}_{25}\times\mathbb{Z}_{25}$. Using the order of elements in $\mathbb{Z}_{100}\times\mathbb{Z}_{...
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185 views

Any divisible subgroup of $G$ splits with internal direct product.

The question is as follows: Let $G$ be an Abelian group and suppose that $A$ is a divisible subgroup of finite index. Show that $G = A \dot{\times} B$ for some $B \leq G$. $\textbf{Some definitions ...
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294 views

Show that $G$ is Abelian if and only if $f: G\times G \to G$ is a homomorphism.

Let $G$ be a group. Let $H$=$G\times G$ be the direct product of $G$ with itself. Define $f: H\to G$ to be $f((g,h))=gh$ for any $(g,h)\in H$. Show that $G$ is Abelian if and only if $f$ is a ...
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120 views

A question on direct summand subgroups

Let $G$ be an abelian group such that $G$ has no nontrivial direct summand subgroup. Is there a characterization of $G$?
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124 views

Why do mathematicians study elementary abelian groups?

I took two algebra courses that I liked as an undergraduate mathematics major in college, but we never covered elementary abelian groups. I recently got interested in the properties of a group I ...
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156 views

$\gcd(|G|, |\text{Aut}(G)|)=1$ means G is abelian?

Prove the following assuming that $G$ is finite group with $\gcd(|G|, |\text{Aut}(G)|)=1$. a) G is abelian (done). b) Every Sylow subgroup of $G$ is cyclic of prime order. Since G is ...
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58 views

Automorphisms of Abelian groups

Let $A$ be a free Abelian group and $N$ a characteristic subgroup of $A$ such that $A/N$ is finite. I also know that $Aut(A/N)$ and $Aut(N)$ are both finite. I have to prove that $Aut(A)$ is finite. ...
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386 views

Commutativity of direct and inverse limits

In exercise 5.34(iv) of Homological Algebra book by Rotman one is asked to prove that direct limits and inverse limits do not necessarily commute. I have two questions : 1.) Is it true that $\...
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1k views

$G$ fin ab group, acts faithfully, transitively on $X$, then $|X|=|G|$

Let $G$ be a finite abelian group. Suppose that $G$ acts faithfully and transitively on a set $X$. Show that $|X|=|G|$. Deduce that the action is equivalent to the action of $G$ on itself by left ...
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49 views

Hungerford Chapter 2 Section 2 Problem 2 WITHOUT using the structure theorem of finite abelian groups

Let $G$ be a finite abelian group and $x$ an element of maximal order. Show that$\langle x \rangle$is a direct summand of $G$. Use this to obtain another proof of Theorem 2.1. Theorem 2.1: Every ...
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144 views

Torsion-free abelian groups, tensor product and $p$-adic integers

I'm studying torsion-free abelian groups and I know (see Fuchs, "Infinite Abelian Groups", vol. $2$, pp $154$) that, if $\mathbb{Z}_p$ is the set of $p$- adic integers and $\mathbb{Z}_{(p)}$ denotes ...
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48 views

Are $\mathbb{Z}$ and $\mathbb{Z}_n$ the only rings (with identity) whose modules are equivalent to abelian groups?

Let $R$ be a ring with identity. Let $M$ and $N$ be $R$-modules. Let $f$ be an (arbitrary) group homomorphism from $M$ to $N$. Under what conditions on $R$,$M$, and $N$ is $f$ also a $R$-module ...
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126 views

Direct sum of subgroups of $\mathbb{Q}^n$ with $\mathbb{Q}$ is a Borel map - Self study

Let $\mathcal{Pow}(\mathbb{Q}^n)$ be the power set of $\mathbb{Q}^n$ and consider the product topology induced by the natural bijection $\mathcal{Pow}(\mathbb{Q}^n)\cong 2^{\mathbb{Q}^n}$ defined by $...
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86 views

On a characterization of abelian groups $G$ based on special commutator relations ($\exists n\in\Bbb N$ s.t. $[x^n,y]=[x,y^{n+1}],\forall x,y \in G$).

Let $G$ be a group. If $\exists n\in \mathbb N$ such that $x^n=yx^n(y^{n+1}x)^{-1}xy^n,\forall x,y\in G$, then how to prove that $G$ is abelian? Thoughts: the condition is same as saying $[x^n,y]=[x,...
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136 views

Two short exact sequences of abelian groups

This is an exercise from Algebraic Topology book of Hatcher: Exercise 2.1.14, pg. 132: Determine whether there is a short exact sequence $ 0 \rightarrow \mathbb{Z}_4 \xrightarrow{f} \mathbb{Z}_8 \...
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61 views

Some nice basis for a subgroup of $Z^k$

I am currently reading the book "Algebra" by Hungerford. I saw some interesting theorem: If $F$ is a free abelian group of finite rank $n$ and $G$ is a nonzero subgroup of $F$, then there exists a ...
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67 views

Invariant factor decomposition of quotient group of two subgroups of $\mathbb{Z}^n$.

Determine the rank and the elementary divisors of the following group: $A/H$ with $A \subset \mathbb{Z}^5$ the group of all $5-$tuples with sum $0$ and $H = A \cap B(\mathbb{Z}^5)$ where $$ B=\begin{...
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827 views

Prove a group of order 28 with a normal subgroup of order 4 is abelian without Sylow Theorems

I have made some headway with the proof, but I can't quite finish it off. Please could I have some help? Please note that at no point are Sylow Theorems to be used during this proof. Let $G$ be a ...
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184 views

Finding 2 nonabelian, nonisomorphic groups of order 225

I need to find 2 nonabelian, nonisomorphic groups of order 225. Here's what I have so far: Let $G$ be a group of order 225. By Sylow's theorems, we have that $G$ contains $P_{25}$ and $P_{9}$, ...
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124 views

Fraction field of group ring of field over torsion free abelian group

Let $G$ be a torsion-free abelian group. If $k$ is a field, it is known that $k[G]$ is an integral domain. Let $k(G)=\operatorname{Frac} k[G]$. If $G,H$ are torsion free abelian groups such that $k(G) ...
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101 views

Concentrated chain complexes

Consider functors $H_n$ and $K(-, n)$, $n>1$. $$\mathbb{Z}\text{Ch} \leftrightarrows \text{Ab}. $$ And take a group morphism $G \to Q$. I would like to compute the Puppe Sequence of $K(G,n) \to ...
3
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64 views

Tensor product with Prüfer $p$-group

Consider the Prüfer $p$-group $$\mathbb{Z}/p^{\infty} = \varinjlim\limits_n \mathbb{Z}/p^n \mathbb{Z}.$$ Let $A$ be any abelian group. Then $A \otimes \mathbb{Z}/p^{\infty}$ is a divisible $p$-torsion ...
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74 views

Counting subgroups of $\mathbb{Z}^3$ with a certain quotient

$ \renewcommand\ZZ{\mathbb{Z}} \renewcommand\calX{\mathcal{X}} \renewcommand\diag{\operatorname{diag}} $Compute the number of subgroups $L \subseteq \ZZ^3$ with the property that $\ZZ^3/L \simeq \ZZ/3\...
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198 views

Does the additive structure of a ring of S-integers detect the ring structure?

Suppose that $K$ and $L$ are number fields with rings of integers $\mathcal O_K$ and $\mathcal O_L$. Let $S$ and $T$ be finite sets of absolute values containing all archimedean valuations of $K$ and $...
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73 views

Decomposition of quotient group of lattices

By the Chinese remainder theorem, we know that $\mathbb{Z}_m \cong \prod_{i=1}^l \mathbb{Z}_{p_i^{k_i}}$, where $m=p_1^{k_1} ... p_l^{k_l}$. Now, let $\Lambda = A(\mathbb{Z}^n) \subseteq \mathbb{Z}^...
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148 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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44 views

Obstruction to be conjugated by an automorphism for subgroups of an abelian group

Let $A$ be a finite abelian p-group( p being a prime number). Let $M,N$ be subgroup such that $M \simeq N$ and $A/M \simeq A/N$ as groups. Can I conclude that there is $\phi \in Aut(A)$ such that $\...
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62 views

Which abelian groups have only a single composition series?

Cyclic groups of composite powers don't: for example, $1=C_1\triangleleft C_3\triangleleft C_6 $ and $1=C_1\triangleleft C_2\triangleleft C_6 $ are both composition series for $C_6$. But cyclic ...
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51 views

Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show that ...
3
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311 views

Rank of an abelian group

I learned that a rank of an abelian group is defined by a cardinality of maximal linearly independent sets. But how we can say that this is well-defined? I mean, I want to show that if $M$ and $N$ ...
3
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142 views

Pisier's $\epsilon$-net condition

I'm reading a book about Sidon sets and I'm stuck on the following proof. In order to facilitate the comprehension of my problems I will give the full proof and the context. Let $G$ be a compact ...
3
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122 views

Power series modulo polynomials

I apologize for the lengthy introduction. It is mainly for context and to introduce a certain phenomenon. $\newcommand{\Z}{\mathbb{Z}}$ Consider the groups $\Z[[x]]$ of formal power series and $\Z[x]$...
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5k views

Find order of given factor group

I'm trying to find the order of this factor group: $$(\mathbb Z_{12}\times\mathbb Z_{18}) / \langle (4,3)\rangle.$$ The order of the factor group is just the number of elements in it (aka the ...
3
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255 views

Partition of Symmetric Group

For symmetric group $S_n$, we need to find a collection of subgroups $G_i$'s such that union of these subgroups is the group $S_n$ and each subgroup found is isomorphic to direct product of cyclic ...
3
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524 views

Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...