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

6
votes
0answers
78 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
votes
0answers
237 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
57 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
53 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
119 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 ...
5
votes
0answers
44 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
160 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 ...
5
votes
0answers
123 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
102 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
37 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
70 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
42 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 ...
3
votes
0answers
31 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 ...
3
votes
0answers
39 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 ...
3
votes
0answers
58 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
votes
0answers
102 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
votes
0answers
68 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 ...
3
votes
0answers
66 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
50 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
118 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
85 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
274 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
36 views

Finding a property for $G/Z(G)$ where $G$ is a nonabelian group

If $G$ is non-abelian group and $Z(G)$ is it's center, what is the least property for $G$ such that $\frac{G}{Z(G)}$ is abelian?
2
votes
0answers
62 views

$gcd(|G|, |Aut(G)|)=1$ means G is abelian?

Prove the following assuming that G is finite group with $gcd(|G|, |Aut(G)|)=1$ a)G is abelian (done) b) Every Sylow subgroup of G is cyclic of prime order. G is abelian than every sylow unique, ...
2
votes
0answers
54 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 $\mathrm{Cong}(X)$ ...
2
votes
0answers
56 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
25 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
358 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 ...
2
votes
0answers
87 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
66 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
65 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
78 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
58 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
166 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
38 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
359 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
108 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
80 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
92 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
144 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
65 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
181 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
14 views

Folner sequence of $\mathbb{Z}[\frac{1}{2}]$

Consider $\mathbb{Z}[\frac{1}{2}]$ consisting of rational numbers of the form $k2^l$ with $k,l\in\mathbb{Z}$. Under addition and discrete topology $\mathbb{Z}[\frac{1}{2}]$ is a discrete abelian ...
1
vote
0answers
41 views

Direct sum notation

I was reading the direct sum of the groups and the index notation looks little bit strange for me. Group $G = \bigoplus_{\alpha < \beta}\mathbb Zx_{\alpha},$ where $\beta$ be an ordinal. Is this ...
1
vote
0answers
37 views

Invariant factors of a subgroup of a subgroup of $\mathbb{Z}^2$

Consider groups $B\leq A\leq\mathbb{Z}^2$. We have: A basis $e_1,e_2$ of $\mathbb{Z}^2$ and integers $a_1,a_2$ such that $a_1e_1,a_2e_2$ is a basis for $A$, and $a_1\mid a_2$. A basis $f_1,f_2$ of ...
1
vote
0answers
24 views

Naively showing that $A_n$ mod a nontrivial normal subgroup is abelian.

Suppose $H \lhd A_n$ is a nontrivial normal subgroup of the alternating group on $n$ letters. Without using the fact that $A_n$ is simple, prove that $A_n/H$ is abelian. Can this be done? I will ...
1
vote
0answers
38 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
85 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
33 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: ...