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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The center of a group is an abelian subgroup

Let $(G,\circ)$ be a group and let $Z(G):=\{x \in G : ax=xa \ \forall \ a \in G\}$ be the center of $G$. How can I show that $Z(G)$ is an abelian subgroup of $G$? What I did so far: $Z(G)$ is a ...
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3answers
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Submodules of a module with a given property

I am curious about the submodules of a module with a given property. Let $M$ be an $R$-module. If $M$ is a finitely generated are the submodules of $M$ finitely generated? If $R=\mathbb Z$, $M$ ...
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1answer
43 views

Number of Nonisomorphic Subgroups of Finite Abelian Group

Lets say I have an abelian group $G$ with order $n$ and I am given the primary components of $G$ and their type. How can I determine how many nonisomorphic subgroups of $G$ there are? And as an ...
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28 views

Let G be a finite Abelian group of order $p^nm$, where p is a prime that does not divide m. Then $G=H\times K$ where H and K are the following sets.

I'm trying to follow this proof in my textbook, Contemporary Abstract Algebra by Gallian (p231) but I'm having trouble understanding what's going on. He writes Let G be a finite Abelian group ...
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0answers
68 views

Questions about completion chapter in Atiyah-Macdonald

I was reading the completion chapter of Atiyah-Macdonald. I have the following questions: (i) What is the topology in the completion group of the topological abelian group? I saw an answer here. But ...
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1answer
72 views

How do I prove this seemingly obvious property of subgroups

The statement is the following: Given an abelian group $G=\langle a_1,...,a_t\rangle$, and a subgroup $H$ of $G$, we need at most $t$ elements to generate $H$; i.e. $H=\langle b_1,...,b_t\rangle$ for ...
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Deduce that the number of inequivalent degree $1$ complex representations of $G$ are equal to $|G|$.

Describe all the one-dimensional complex representations of a finite abelian group $G$. Deduce that the number of inequivalent degree $1$ complex representations of $G$ are equal to $|G|$. attempt: ...
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33 views

Abstract Algebra Elementary Properties of Groups

This is Excercise 4.A.5 from Pinter's "A Book of Abstract Algebra": Let $a$, and $x$ be elements of a group $G$. Solve for $x$ in terms of $a$. Solve Simultaneously: $x^2 = a^2$ and $x^5 = e$ ...
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71 views

When are groups subgroups of a same group?

Assume that $G_{1}$ and $G_{2}$ are groups and that $G_{1} \cap G_{2}$ has a group structure that makes it a common subgroup of $G_{1}$ and $G_{2}$. In other words, the set $G_{1} \cap G_{2}$ is a ...
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If $ G = \{g_1, g_2, …, g_n\}$ is a finite abelian group, then for any $x \in G$, $xg_1 \cdot xg_2 \cdots xg_n = g_1 \cdot g_2 \cdots g_n$

Let $G = \{g_1, g_2, \dots, g_n\}$ be a finite abelian group, prove that for any $x ∈ G$, the product $$xg_1 \cdot xg_2 \cdot \cdot \cdot xg_n = g_1 \cdot g_2 \cdot \cdot \cdot g_n.$$ I can ...
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Isomorphism between $G$ and $\mathbb{Q}^{*}$

Let $\{G_{n}\}_{n\in \mathbb{N}}$ be a family of additive groups with $G_{1}=\mathbb{Z}_{2}$ and $G_{n}=\mathbb{Z}$ for $n\geq 2$ $$G=\bigoplus_{n\in \mathbb{N}}G_{n}$$ I want to prove that $G\cong ...
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1answer
58 views

Are all countable torsion-free abelian groups without elements of infinite height free?

The height of an element $g$ in an abelian group $G$ is the largest $n\in \mathbb{N}$ such that there exist an element $h\in G$ such that $n*h=g$. If $g$ has no such largest integer than $g$ is of ...
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3answers
65 views

A line avoiding an Algebraic group

Let $K$ be an algebraically closed field, and $G\subset (K,+)^3$ an algebraic subgroup (i.e. given as the zero sets of finitely many polynomial equations) of dimension 1. Is it clear that there is a ...
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44 views

Compact abelian group and Continuous functions

If $G$ is a compact abelian group, $\widehat{G}$ is the dual group of $G$,i.e. all the continuous homomorphism from $G$ to $S^1$,$S^1=\{z\in \mathbb{C}\big | |z|=1\}$. Show that the linear span of ...
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65 views

On algebraic groups of dimension 1

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let $K$ be an algebraically closed field. ...
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1answer
17 views

Continuous functions on compact group and uniformity

If $G$ is a compact abelian group and $f\in C(G)$. Then $\forall \epsilon >0$,there exists an open neighbourhood $U$ of $0\in G$, such that $\forall g\in G , \forall u_1,u_2\in U$, we have ...
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43 views

Abelian subgroup of standard wreath product

Let $A$ and $B$ be non-trivial groups. We construct their (restricted) wreath product as follows. Denote by $A^{(B)}$ the set of all function from $B$ to $A$ with finite support, and equip it with ...
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1answer
50 views

Quotient of direct sum of abelian groups [closed]

Let $A \oplus B \simeq A' \oplus B $. Does it follow that $A\simeq A'$? Many thanks in advance!
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Showing a non-isomorphism of groups

I need to show that $\Bbb Z^*_8$ is not isomorphic to $\Bbb Z^*_{10}$. $\Bbb Z^*_n$ means integers up to $n$ coprime with $n$ I do not know how to do this. I have difficulties doing proofs ...
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30 views

Which one of the below options is correct?

I think the option $(Q)$ is true since $O(Q/\{-1,1\})= 8/2 = 4 = 2^2$. Since order is $p^2$ thus $(Q)$ option is true. Can anyone suggest about option $(P)$? Thanks
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1answer
22 views

Several true/false statements about a finite group $a,g\in G$ such that $a$ is of order $2$

Let $G$ be a finite group, and $a,g\in G$ such that $a$ is of order $2$, then the following is either true or false: The element $gag^{-1}$ is of order $2$. $(ag)^2=g^2$ if $ag$ is of ...
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1answer
41 views

Characteristic subgroups of order $2$

Could anybody give an example of a finite abelian $2$-group with more than one characteristic subgroup of order $2$ ? (In other words, a finite abelian $2$-group $G$ with a Klein subgroup $V$ such ...
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26 views

Abelianizated free product of two groups

Given $$G=\mathbb{Z}_2*\mathbb{Z}_2=P(a,b\mid a^2,b^2)$$ among other things I wanted to show that this group is infinite, what I did is consider the words of the form $$abababa\ldots$$ they are all ...
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87 views

Does the determinant give you the index over $\mathcal{O}_k$ as well as over $\mathbb{Z}$?

It is a standard fact that if $M$ is a nonsingular $n\times n$ integer matrix, the index of the $\mathbb{Z}$-span of its columns as an abelian group in $\mathbb{Z}^n$ is $|\det M|$. What happens if we ...
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81 views

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ?

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ? ( I know that there 'is' a 'surjection' , but I don't know whether any surjective homomrophism from $\mathbb R$ ...
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25 views

freeness of vector spaces and abelian groups

This question is continuation of my previous question Extension of vector spaces and abelian groups Given a diagram of linear transformations of $K$ vector spaces $$B\xrightarrow{\epsilon} ...
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1answer
69 views

Where does the group $\mathbb Z/(a)\oplus \mathbb Z/(a^2)\oplus \cdots $ arise?

Let $a>1$ be an integer, and consider the infinite abelian group $$ V_a=\bigoplus_{j=1}^{\infty}\mathbb Z/{a^j\mathbb Z}. $$ Can anyone provide references to places where this (or related) groups ...
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Extension of vector spaces and abelian groups

While reading about modules from Hilton & Stammbach's Homological algebra, I saw the following statement : $\Lambda$ is a ring. $\Lambda$ modules are generalizations of vector spaces and ...
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1answer
93 views

Subgroups of finite index of $\mathbb Z/2\mathbb Z\times \mathbb Z$

Let $H$ be a subgroup of index $n$ in $(\mathbb Z/2\mathbb Z)\times \mathbb Z.$ Is there finitely many subgroups of finite index of $(\mathbb Z/2\mathbb Z)\times \mathbb Z$ ? If yes, can we ...
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26 views

Baer-Specker group versus free abelian group generated by an uncountable set [duplicate]

I just learned on Wikipedia that the Baer-Specker group, that is, the group of all integer sequences, is not free abelian. I'm hoping I could be helped to understand why this is true by someone ...
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1answer
59 views

Prove that (possibly) infinite group of all invertible maps of X to itself is not Abelian.

I have this question on my assignment and I this fact seems trivial to me, but I can not come up with a rigorous proof. I thought to go by contradiction: Assume such a group $G$ is Abelian -> ...
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1answer
26 views

Showing that $\mathbb{C}^\times$ is an abelian group

QUESTION Multiplication of complex numbers defines a binary operation on $\mathbb{C}^\times := \mathbb{C} \setminus \{0\}$. Show that $\mathbb{C}^\times$ together with this operation is an abelian ...
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1answer
29 views

Abelianization of $\mathbb{Z}_2*\mathbb{Z}_3$

Intuitively it has to be $$\text{Ab}(\mathbb{Z}_2*\mathbb{Z}_3)=\mathbb{Z}_2\times\mathbb{Z}_3$$ here is my approach on how to prove it $$\mathbb{Z}_2=P(a\mid a^2),\mathbb{Z}_3=P(b\mid b^3)\Rightarrow ...
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1answer
36 views

Find the structure of $\mathbb{Z}_{120}^*$

How to find the structure (in term of cyclic groups) of $\mathbb{Z}_{120}^*$? I know that the number of elements of $\mathbb{Z}_{120}^*= \phi(120) = 32 = 2^5$ But then, any hints?
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1answer
29 views

Complements of torsion-free abelian groups [closed]

Here is my question: Let A be an abelian torsion-free group, and B be a subgroup of A. Can we always find another subgroup C of A such that A/B is isomorphic to C?
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1answer
38 views

Subgroups order $p$ in a non-cyclic abelian finite p-group.

Is it true that if $G$ is a finite abelian non-cyclic $p$-group then a subgroup of order $p$ cannot be unique? How can I prove it if the sentence is correct? Excuse me for the question, but I've some ...
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1answer
34 views

A question on finitely generated Abelian groups with a minimal number of generators

In my class on group theory I have encountered this strange looking question relating to Abelian groups in terms of generators which states: We are to find, up to isomorphism, all Abelian groups ...
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1answer
22 views

Do the monoids (under integer multiplication) $\mathbb{Z}/_{51}$ and $\mathbb{Z}/_{15}\oplus\mathbb{Z}/_{5}$ have isomorphic groups of units?

This is one of my assignment using fundamental theorem of finitely generated abelian groups. However, I don't really know how to find the smallest generating sets of ${(\mathbb{Z}/_{51})}^\times$ and ...
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3answers
89 views

Subgroup of $\mathbb{Q}$ containing $\mathbb{Z}$

Question: Among all the subgroups of $\mathbb{Q}$ containing $\mathbb{Z}$, does there exists a maximal (proper) subgroup? I have proved these facts: (1) $\mathbb{Q}$ has no maximal (proper) ...
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1answer
13 views

Invariants of a finite abelian group written as a central extension of a cyclic group by a finite abelian group.

Notation : If $A$ is a finite abelian group then $(d_r,...,d_1)$ are the invariants of $A$ if $d_r>1$ : $$A\text{ is isomorphic to } \mathbb{Z}/d_r\times...\times \mathbb{Z}/d_1 \text{ and } ...
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1answer
23 views

Semidirect product of two groups defined in terms of a homomorphism

I am going through From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups by Bacon et al. I am having trouble to understand the ...
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192 views

Prove that middle cancellation implies that the group is abelian

Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian. (Middle cancellation implies commutativity). ...
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1answer
40 views

Abelian Group as the quotient of a free Abelian Group

Is it true that every abelian group $G$ is the quotient of a free abelian group $F$? I think so, since every abelian group $G$ is the quotient of a free group $H$ under some relations, but some of ...
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1answer
54 views

All elements of this finite abelian group.

$$A=\left(\begin{matrix}1 & 2 & 2 \\ 2 &2&2\\3&4&2 \end{matrix}\right)$$ Let $H$ be a subgroup of $\mathbb{Z}^3$ generated by the vectors $\vec{g_i} = \sum_{j=1}^3 ...
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1answer
66 views

The order of a non-abelian group is $pq$ such that $p<q$. Show that $p\mid q-1$ (without Sylow's theorem) [duplicate]

The order of a non-abelian group is $pq$ where $p$ and $q$ are primes such that $p<q$. Show that $p\mid q-1$ (without anything to do with Sylow's theorem). How to start? I tried already some ...
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1answer
45 views

Subgroups of $Z^n$ are finitely generated

I have read a couple of proofs already, but all of them try to go further and start talking about modules. Is there any more direct proof of this fact without using modules?
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21 views

Determine which abelian groups can be the central term of this exact short sequence

I am trying to solve the following problem: Determine which abelian groups $A$ can appear as central terms in a short exact sequence $\mathbb{Z} \to A \to \mathbb{Z} \oplus \mathbb{Z}_5$ What ...
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17 views

A group with unusual discrete log properties.

Does there exist a group where computing $g^x$ from $g^{a^{x}}$ is easy, computing $g^{a^{x}}$ from $x$ and $g^{a}$ is hard, and computing $x$ from $g^a$ and $g^{a^x}$ is hard. Intuitively I would ...
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34 views

Meaning of abelian subquotient

I was reading an article, somewhere it says that the "abelian subquotients of the group $G$" are .... How does it defined ? For example if we take $G=S_n$, the symmetric group, then what are the ...
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62 views

$(G,+)$ abelian group is divisible $\Longleftrightarrow$ it's an homomorphic image of $\Bbb Q^{(X)}$

Let $(G,+)$ be an additive abelian group. Let us suppose $G$ divisible (i.e. $G=nG\;\;\;\forall n\ge1$). Let then $x,y\in G$. Then there exists $z\in G$ and $n,m\ge1$ such that $x=nz$ and ...