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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Regular semigroups- normal semigroups!

If I take $S$ to be a Clifford semigroup with the set of idempotents $E$, then $S'$ let be a semilattice with the same set of idempotents( $E$) such that for every $e \in E$, $S_{e}$ is a normal ...
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48 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
10 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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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
44 views

Quotient of direct sum of abelian groups [on hold]

Let $A \oplus B \simeq A' \oplus B $. Does it follow that $A\simeq A'$? Many thanks in advance!
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45 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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28 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
36 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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23 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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65 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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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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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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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
60 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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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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34 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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156 views

Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
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1answer
47 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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27 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
74 views

Orders of the elements in $\mathbb{Z}/8\mathbb{Z}$

I know that the order of an element $a$ in a group is the smallest positive integer $n$ such that $a^n = 1$. You know $\mathbb{Z}/8\mathbb{Z} = \{\overline{0}, \overline{1}, \dotsc, ...
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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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136 views

Show that $G\{m\}$ is finite for all non-zero $m\in \mathbb{Z}$

Let $G$ be an abelian group, and let $m$ be an integer, then we define $G\{m\} := \{a\in G:ma=0_G\}$. Now, suppose that $G$ is an abelian group that satisfies the following properties: (i) For all ...
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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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Cubic Planar Graphs have $2^m-1$ Hamilton Cycles, contradicting Bosak…

I looked at the symmetric difference of hamilton cycle (HC) in cubic planar graphs and found that, together with the empty graph, they build a subgroup of the abelian group $\Omega$ of symmetric ...
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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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1answer
76 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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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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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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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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735 views

Quotient of two free abelian groups of the same rank is finite?

Let $A,B$ be abelian groups such that $B\subseteq A$ and $A,B$ both are free of rank $n$. I want to show that $|A/B|$ is finite, or equivalently that $[A:B ]$ (the index of $B$ in $A$) is finite. For ...
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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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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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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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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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$(G,+)$ 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
39 views

Universality of tensor product from monoidal structure

As a follow-up to this previous question of mine, I'm trying to understand how to obtain tensor products from internal homs. I'm having a lot of difficulties and have found myself stuck already in ...
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1answer
37 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
72 views

Example of a non-abelain finite group $G$ with $G/N$ abelian and infinite group $G$ with $G/N$ finite

Have not been able to think of a examples with the following properties: Example of a non-abelian finite group $G$ with property that $G/N$ is abelian for every non-trivial normal subgroup $N$ of ...
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201 views

Finite Abelian groups with the same number of elements for all orders are isomorphic

Let $A$ and $B$ be finite abelian groups. Suppose that for every natural number $m$, the number of elements of order $m$ in $A$ is equal to the number of elements of order $m$ in $B$. Prove that $A$ ...
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1answer
25 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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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
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 ...