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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36 views

What are the invariant factors of $(\mathbb{Z}/(1000))^\times$?

I'm curious about the invariant factors of $(\mathbb{Z}/(1000))^\times$. I put down $$ (\mathbb{Z}/(1000))^\times\cong(\mathbb{Z}/(8))^\times\oplus(\mathbb{Z}/(125))^\times $$ It's easy to compute by ...
5
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101 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, ...
5
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52 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). ...
5
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100 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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42 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 ...
4
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114 views

$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
4
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193 views

Characterization of the First Order Theory of Ordered Abelian Groups via Quantifier Elimination

In the paper "Elimination of Quantifiers in Algebraic Structures" Macintyre, McKenna and van den Dries, proved that every field (ordered field) whose theory admits quantifier elimination in the ...
3
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36 views

Definition of pure-essential extension

Let $B$ be a pure subgroup of an abelian group $A$. In his book "Infinite Abelian Groups", Academic Press, 2vols., Fuchs defines $A$ to be a pure-essential extension of $B$ if there is no nonzero ...
3
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78 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 ...
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82 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 ...
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23 views

Counting homomorphisms by the order of their images

I am trying to count homomorphisms from $\mathbb Z^r$ to $(\mathbb Z/m)^n$ while keeping track of the order of the image of each map. In other words, for each integer $k$ dividing $m^n$, I want to ...
2
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59 views

Epimorphism (abelian group)

Let $(G,\cdot), (H,*)$ Groups and $f: G\rightarrow H$ an Epimorphism. Show that: If G is an abelian group, then H is also an abelian group. Is the reversal of this proposition also true? My idea: ...
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52 views

Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
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48 views

Abelian SubGroup Variant:

Consider the following problem: Find integers $x_1, x_2, x_3,\dots, x_n$ Such that: $$P(x_1,x_2,\dots, x_n) = Q$$ for some integer $Q$ and polynomial $P$ where for all permutations of any set of ...
2
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68 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 ...
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54 views

Abelian group $G$ such that $pG=p^2G$ for a fixed prime $p$

I know that an abelian group $G$ such that $pG=p^2G$ is of the form $G=D(G)\oplus X$, where $D(G)$ is the maximal divisible subgroup of $G$, $D(X)=0$, $X/T(X)$ is divisible and $pX_p=0$, where $X_p ...
2
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150 views

Cardinality relation between subsets of a group

$G$ is an abelian group, $A$ and $B$ are non empty finite subsets of $G$. Set $A+B := \{a+b\mid a\in A, b\in B\}$ and $H := \mathrm{stab}(A+B)=\{g\in G \mid g+A+B = A+B\}$. Prove that $$ ...
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36 views

Separable elements of a finite abelian group

Let $\mathbf G$ be a finite abelian group, let $a, b \in \mathbf G$, and let $\langle a \rangle$ and $\langle b \rangle$ be the cyclic subgroups of $\mathbf G$ generated by $a$ and $b$ respectively. ...
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180 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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95 views

Distributions over locally compact Abelian groups: when can they be Fourier transformed?

Pontryagin duality shows us every locally compact Abelian group---such as $\mathbb{R}^n$, $\mathbb{Z}$, the circle $\mathbb{R}/\mathbb{Z}$ or any finite Abelian group---has a Fourier transform. In the ...
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66 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 ...
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76 views

Asking about Ulm invariant of abelian group $G$

I am backing to read my previous question and learn more facts which the Masters left for me within comments. One of them appeared here Verifing some properties about $G$. There; I was verifying that ...
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83 views

Isomorphism of annihilator of a subgroup in the context of group characters

I am trying to learn about characters of finite abelian groups. A character is a homomorphism from a finite abelian group $G$ into the multiplicative group of complex numbers of absolute value 1. In ...
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199 views

Direct sum of Prüfer groups and $\mathbb Q/\mathbb Z$

It can be easily shown that, the Prüfer $p$-group $\mathbb Z(p^\infty)$ is isomorphic to multiplicative group $$R_p=\{e^{2\pi ik/p^n}|k\in\mathbb Z,n\geq0\}$$ Now I want to prove that: ...
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137 views

Classification Theorem Of Abelian Groups-Question regarding the proof

I'm currently reading Munkres-Algebraic topology text, and in his review chapter of abelian groups, he gives the classification theorem for finitely generated abelian groups. He ommited some important ...
2
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62 views

having a subgroup of $\mathbb{Z}^{3}$ and wants to show linear independency with parameters

I am stuck with these hard-star exercises for some time now (they are from a book called "Introduction a L'Algebre et L'Analyse Modernes" de M.Zamansky"), if somebody sees the right way, I will be ...
2
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0answers
174 views

A question about reduced torsion abelian groups

If a reduced torsion abelian group has no cyclic direct summands of order greater than 2, is it an elementary abelian 2-group? Background: I'm trying to classify the groups whose group rings have a ...
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23 views

Automorphism group of an abelian p-group

I'd like to know if it's known the structure of the automorphism group of an abelian $p$-group with the minimal condition on subgroups, for some prime number p. I know that if $A$ is an abelian ...
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16 views

list of conjugacy classes in elementary abelian p-group

Let G be an elementary abelian p-group, how can I get a complete list of conjugacy classes in G? A general structure of the conjugacy classes will do. Thank you in advance. Magero Fidelius
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68 views

Prove that a(mn)=a(m)a(n), (n,m)=1

Given a positive integer $n$ where $a(n)$ is the number of non-isomorphic abelian groups of order n. 1) Prove that $a(mn)=a(m)a(n), (n,m)=1$ 2) Prove that $a(p^k)$ is the number of partitions of k, ...
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61 views

Working with special cases of the converse of Lagrange's theorem

I am to answer true/false to statements on the form: Every abelian group of order divisible by $n$ contains a cyclic subgroup of order $n$. This follows directly from Cauchy's theorem when $n$ ...
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47 views

Find which of the abelian groups are isomorphic to $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}_{6},\mathbb{Q}\oplus \mathbb{Z}_{3})$

Which of the abelian groups are isomorphic to $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}_{6},\mathbb{Q}\oplus \mathbb{Z}_{3})$?
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122 views

Isomorphism theorem for Abelian groups, related to Hatcher exercise 2.1.14

I am trying to understand which Abelian groups can fit the short exact sequence \begin{equation} 0 \rightarrow \mathbb{Z}_{p ^m}\rightarrow A \rightarrow \mathbb{Z}_{p^n}\rightarrow 0. \end{equation} ...
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38 views

Question about finitely generated abelian groups

Why is an Abelian group finitely generated iff $A/mA$ is finite for some $m\gt 1$ and $A$ has a norm function? I know that $mx$ where $x$ is an element of $A$ is equivalent to $0$ in $A/mA$, and I ...
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83 views

Orders of Elements in Minimal Generating sets of Abelian p-Groups

I'm looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders ...
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60 views

Specific basis of a subgroup of a free abelian group

I'm looking for clarification on Fraleigh's "A First Course in Abstract Algebra" Theorem 38.11. It states: "Let $G$ be a nonzero free abelian group of finite rank $n$, and let $K$ be a nonzero group ...
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53 views

Order of elements of a group.

Assume that G is an abelian group, and a∈G. (a) Assume that |a|=r and that m|r, say r=mt. Prove that $|a^t|$=m. Proof:Assume a∈G and |a|=r and m|r, say r=mt. Assume $|a^t|$=k Since ...
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49 views

If a commutative group has subgroups of order $m$ and $n$ , then it has a subgroup of order $\space$l.c.m.$(m,n)$

Without using ideas from "Normal subgroups" and "Quotient groups" how do we prove the following statement ; "If a commutative group has subgroups of order $m$ and $n$ , then it has a subgroup of order ...
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80 views

Structure of finite abelian group

I am trying to solve this problem, and I think I should use the structure theorem for finite abelian groups, but i can't really figure this out. Let $G$ be a finite abelian group. Prove there is a ...
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44 views

Product and quotient in Abelian groups

In linear algebra, for any vector space $V$ and its subspace $U$, there exists a subspace $W$ of $V$ such that $U\oplus W=V$ and $W\cong V/U$. Does similar property hold for Abelian groups? That is, ...
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57 views

Finding subgroups to special groups

Let $G$ and $H$ be groups. Is there any possibility to find all (normal) subgroups of $G\times H$ and $G*H$? I really hope that this task is easier, if $G$ and $H$ are cyclic groups. I tried to find ...
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150 views

Rubik cube theory and commutative matrices

I was reading the same paper on solving the Rubik cube. The 20 pages or so are mostly proofs and introduction to group theory which preface an algorithm for solving near the end. On pages 13-14 they ...
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51 views

How do I compute this abelian group?

If $G = \bigoplus_i \mathbb{Z} / \langle w_i\rangle_i $ is a finitely generated abelian group and $w_i = \sum_j m_{ij} v_j$ such that $\{v_j\}_j$ is a basis for $\bigoplus_i \mathbb{Z}$, then $G = ...
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85 views

Free and torsion Group

Can you please explain torsion subgroup and free subgroup of free abelian group? and also if $G$ is a finitely generated abelian group; how is $G$ a direct product of free part and torsion part? ...
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196 views

Euler-Poincaré characteristic and homology

$\DeclareMathOperator{rk}{\text{rk}}$ $\DeclareMathOperator{im}{\text{im}}$ The problem Let $$C = ( C_n \overset{\partial_n}\to C_{n-1} \overset{\partial_{n-1}}\to \dots \overset{\partial_2}\to C_1 ...
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72 views

If $G=\prod_{p\in P}\mathbb Z_p$ then $\frac{G}{tG}$ is divisible.

I want to show that: If $G=\prod_{p\in P}\mathbb Z_p$, wherein $P$ is the set of all primes, then $\frac{G}{tG}$ is divisible. I know that $tG$ is not a direct summand and if $x\in G$ wants to ...
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62 views

Pontrjagin's Lemma and an application

I would appreciate any kind of help on the following issue: On page 114 of Rotman's "Homological Algebra", exercise 3.4 reads: 1) (Pontrjagin) If an abelian group $A$ is countable, torsion-free ...
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93 views

Computing a generating set of the kernel of a module

Given a generating set of a $\mathbb{Z}$-module $M \subseteq {\mathbb{Z}_k}^n$, is there a known algorithm to compute a generating set of $\{u \in {\mathbb{Z}_k}^n \, : \, \forall v \in M \quad v ...
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47 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, $G\cong G_t\oplus F_1$ ...
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0answers
15 views

maximal and minimal subgroups of torsion abelian groups

Is there a torsion abelian group $G$ for which exactly one of these conditions is correct: 1) every non-trivial subgroup of $G$ contains a minimal (non-trivial) subgroup of $G$. 2) every proper ...