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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9
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0answers
181 views

Writing this $G$-set explicitly as union of orbits

Let $G$ be a finite abelian group, and let $A$ and $B$ be subgroups. I'm interested in $G/A\times G/B$ with its natural $G$-set structure. In $G/A\times G/B$, the stabilizer of any element is $A\cap ...
6
votes
0answers
208 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 ...
6
votes
0answers
56 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
votes
0answers
39 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 ...
5
votes
0answers
41 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
votes
0answers
105 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
votes
0answers
101 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 ...
4
votes
0answers
33 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. ...
4
votes
0answers
80 views

A question about the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$

I wanted to clarify some confusion I was having on the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$, which I call $Aut(\mathbb{Z}_{2} \times \mathbb{Z}_{4})$. I considered the ...
4
votes
0answers
139 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 ...
3
votes
0answers
50 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 ...
3
votes
0answers
54 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 ...
3
votes
0answers
40 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
votes
0answers
99 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 ...
3
votes
0answers
84 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 ...
3
votes
0answers
247 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: ...
2
votes
0answers
45 views

Classify $\mathbb{Z_6} \times \mathbb{ Z_{24}} / \langle(3,2)\rangle$ according to fundamental theorem of finitely generated abelian group

The order of $G/H = 12$ So it can be isomorphic to $\mathbb{Z_3} \times \mathbb{Z_4}$ or $\mathbb{Z_3} \times \mathbb{Z_2} \times \mathbb{Z_2}$ $(0,1)$ has order of 4, $(1,0)$ has order of 12, ...
2
votes
0answers
24 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
votes
0answers
72 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: ...
2
votes
0answers
58 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 $g \in H$?
2
votes
0answers
63 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 ...
2
votes
0answers
49 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
votes
0answers
70 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 ...
2
votes
0answers
56 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
votes
0answers
153 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 $$ ...
2
votes
0answers
37 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. ...
2
votes
0answers
289 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 ...
2
votes
0answers
100 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 ...
2
votes
0answers
72 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 ...
2
votes
0answers
83 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 ...
2
votes
0answers
107 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 ...
2
votes
0answers
139 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
votes
0answers
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
votes
0answers
178 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 ...
1
vote
0answers
43 views

If $X$ is an Abelian group, then $\ker_X : \mathrm{Cong}(X) \rightarrow \mathrm{Sub}(X)$ is a bijection. Is there a partial converse?

(All monoids are written additively in this question, even the non-commutative ones.) Given a monoid $X$, write $\mathrm{Sub}(X)$ for the lattice of submonoids of $X$, and write $\mathrm{Cong}(X)$ ...
1
vote
0answers
36 views

Automorphisms of $Z_{p^{i_1}}*Z_{p^{i_2}}*…*Z_{p^{i_n}}$

If $Z_{p^{i_1}}\times Z_{p^{i_2}}\times\cdots\times Z_{p^{i_n}}=\langle a_1,...,a_n\rangle$, then each automorphism of this group is the forms as follows, $$\sigma:a_j\rightarrow ...
1
vote
0answers
62 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 ...
1
vote
0answers
88 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
vote
0answers
17 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: ...
1
vote
0answers
34 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
0answers
73 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
0answers
33 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
vote
0answers
37 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 ...
1
vote
0answers
34 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
40 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 ...
1
vote
0answers
33 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 ...
1
vote
0answers
20 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
1
vote
0answers
64 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 $n$ ...
1
vote
0answers
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 $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}_{6},\mathbb{Q}\oplus \mathbb{Z}_{3})$?
1
vote
0answers
171 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} ...