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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71 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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0answers
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
66 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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2answers
40 views

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
90 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
57 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
35 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
34 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
32 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
21 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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2answers
184 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
39 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
61 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 ...
2
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1answer
43 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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1answer
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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0answers
32 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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2answers
60 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 ...
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1answer
33 views

A criterion for isomorphism of finite abelian groups using fundamental theorem of finitely generated abelian groups [duplicate]

I have recently encountered this very interesting problem from my abstract algebra class where we have just now proven the fundamental theorem of finitely generated Abelian groups, and the problem ...
0
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1answer
47 views

The homomorphic image of an abelian group is abelian

We learnt about Group Homomorphisms and Abelian Groups, but never have we been shown how to tackle such question....and I have an exam on this tomorrow. The question says: Let $\phi : G ...
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1answer
58 views

Is every abelian group a product of cyclic groups?

This lecture notes from John Jones https://www2.warwick.ac.uk/fac/sci/maths/people/staff/vincent/cohomology.pdf state that abelian groups are a product of cyclic groups (page 9). We know that this ...
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1answer
41 views

find torsion coefficients of groups

I have to find torsion coefficients of groups $G_1\simeq Z/2\oplus Z/4\oplus Z/3\oplus Z/3\oplus Z/9$ and $G_2\simeq Z/15\oplus Z/20\oplus Z/18$. I want to ask if my calculations are correct. For ...
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1answer
22 views

Invariant factors and elementary divisors of an abelian group

I have to find the elementary divisors and invariant factors of : $$ \mathbb Z_6\oplus\mathbb Z_{20}\oplus\mathbb Z_{36}$$ I'm following this. I think that elementary divisors are ...
3
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1answer
233 views

Intervals in divisible ordered groups

Is it true that if $(G,+,0,<)$ is a divisible ordered abelian group with at least two elements, then for $a,b >0 \in G$, there is an injective order preserving map from $[0;a)$ to $[0;b)$? It ...
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0answers
22 views

Characters of a Finite Abelian group

Let $C_m$ denote the cylic group of order $m$. Recall $\widehat{C}_m=\{ \psi_j: j\in C_m\},$ where $\psi_j$ is defined by \begin{align} \psi_j(k)=e\Big( \frac{2\pi \sqrt{-1}jk}{m}\Big). \end{align} ...
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1answer
37 views

The central product of two cyclic subgroups of prime power order for one $p$ is isomorphic to direct product of two cyclic groups of prime power order

For two groups $G_1, G_2$ and two central subgroups $U_1 \le Z(G_1), U_2 \le Z(G_2)$ which are isomorphic by some given $\mu : U_1 \to U_2$ the central product is the group $$ (G_1 \times G_2) / D $$ ...
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1answer
80 views

Is there a known formula for the “cyclicity” of a positive integer?

Given a positive integer $n$, let us define that the cyclicity of $n$ is the number of multitsets of cyclic numbers (distinct from $1$) whose product is $n$. For example, the cyclicity of $15$ is $2$, ...
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1answer
19 views

Z modules spanned by row space of matrix invariant under matrix multiplication

I have met this strange looking problem on which I have no idea, from my course on Abstract Algebra dealing with modules: Let $ v_1,...,v_k \in \mathbb{Z}^n $ row vectors of length n over $ ...
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4answers
145 views

Why are there $12$ automorphisms of $\Bbb Z\oplus \Bbb Z_{3}$?

Let $A:=\Bbb Z\oplus \Bbb Z_{3}$, then what is $|\text{Aut}(A)|$? My answer is $4$ but the correct answer (without explanation) turns out to be $12$! How come? Well my understanding is, it just ...
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2answers
30 views

A problem in decomposing a p group into direct sum of nontrivial subgroups

Hello all I have taken a group theory course where we are now covering p groups and we I have met the following exercise: Let $ G = Z/(p^n) $ is a(n Abelian) group of order $ p^n $ for a prime $ p ...
5
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1answer
81 views

if $ab = ba$ for all $a \in X$ and for all $b \in X$ then $\langle X \rangle$ is abelian subgroup of $G$

if $X \subseteq G$ such that $\forall a,b \in X$ we have $ab = ba$ then we should prove that $\langle X \rangle$ is an abelian subgroup of G. its abviouse that $\langle X \rangle$ is subgroup of $G$. ...
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2answers
55 views

Is every subgroup of a free abelian group a direct summand?

My guess is NO, because take $G=\mathbb{Z}$ and $F=2\mathbb{Z}$ is a subgroup but not a direct summand.
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1answer
32 views

A problem in Abelian p-group being indecomposable

I have recently met this very interesting problem in my Group theory course: Let $ p $ be a prime number and $ 1 \leq n $ is a natural number such that G is the Abelian p-group $ G = ...
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2answers
85 views

A finite abelian group $A$ is cyclic iff for each $n \in \Bbb{N}$, $\#\{a \in A : na = 0\}\le n$

Let $A$ be a finite abelian group. Prove that $A$ is cyclic iff for each $n \in \Bbb{N}$ $$\#\{a \in A : na = 0\}\le n.$$ Any help or hint will be appreciated.
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68 views

Examples of torsion-free abelian groups with finite automorphism group

$\mathbb{Z}$ is a torsion-free abelian group with finite automorphism group. Are there other examples of such groups? Jumping from $\mathbb{Z}$ to $\mathbb{Q}$ is not good; since $\mathbb{Q}$ has ...
3
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1answer
51 views

Minimal number of generators for a finitely generated abelian $p$-group

Let $A = \text{Tor}_p(A)$ be a finitely generated abelian $p$-group. (Here $p$ is prime). Show that the minimal number of generators of $A$ is $\log_p|A/pA|$. What I tried - I think that from ...
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2answers
73 views

Show that $\mathbb Z_p \oplus \mathbb Z_p \oplus \mathbb Z_p$ is not generated by two elements

Show that group $A = \mathbb Z_p \oplus \mathbb Z_p \oplus \mathbb Z_p$ is not generated by two elements. ($p$ is prime.) any help or hint will be appreciated.
2
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0answers
49 views

Number of constituents in invariant factor decomposition of kernel of homomorphism

Notation. Given a finite abelian group $ G $, the invariant factor decomposition theorem ensures a the existence of $ k_1 \mid \cdots \mid k_n $, all different, such that $ G \simeq \bigoplus_{i=1}^n ...
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1answer
37 views

Example of an endomorphism on an abelian group that is not left multiplication

It is well-known that all endomorphisms on the abelian group ($\Bbb{Z}$,+) can be seen as a left multiplication by some element in some ring structure on ($\Bbb{Z}$,+); namely left multiplication by ...
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4answers
104 views

Is there a non abelian group of order 759? [closed]

I tried to use Sylow theorems to prove that there is not, but it is not trivial.
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3answers
70 views

Number of elements of order $2$ in Abelian groups of order $2^{n}$

I'm self studying some group theory and one of the exercises I came across: Question: Prove that an Abelian group of order $2^{n}, n \in \mathbb{N}$ must have an odd number of elements of order 2. ...
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1answer
72 views

Find the number of elements of order 2 and number of subgroups of index 2.

Let $A= \mathbb Z_{60} \oplus \mathbb Z_{45} \oplus \mathbb Z_{12} \oplus \mathbb Z_{36}$. 1) what is the number of elements in A with order 2. 2) what is the number of subgroups of A ...