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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38 views

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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2answers
22 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
21 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
30 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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0answers
22 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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1answer
60 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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2answers
57 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
58 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
33 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 ...
4
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1answer
71 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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25 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
44 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
26 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
34 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
27 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
30 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
20 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
80 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
11 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
20 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
155 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
34 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
45 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
41 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
18 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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0answers
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
26 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
53 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
24 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 ...
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1answer
36 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 ...
2
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1answer
50 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
26 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
15 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
26 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
20 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
27 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 $$ (G_1 \times G_2) / D $$ ...
5
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1answer
75 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 $2$, ...
2
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1answer
18 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
107 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
28 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
80 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
46 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
29 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 = ...
2
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2answers
82 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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56 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 ...
2
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1answer
47 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
71 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.