# Tagged Questions

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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### $H \subseteq K$ be subgroups of an infinite abelian group $G$ such that $G/H \cong G/K$ , then are $H,K$ equal or atleast isomorphic?

Let $H \subseteq K$ be subgroups of an infinite abelian group $G$ such that $G/H \cong G/K$ , then is it true that $H=K$ ? Or atleast $H \cong K$ ? ( If $G$ were finite then it would be trivially true ...
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### Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups

Let $L\subset \mathbb{Z}$ be the subgroup of $\mathbb{Z}^3$ generated by the elements $(-1,-1,4),(2,4,0),(3,3,8)$. Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups. I've tried creating a ...
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### If $G$ is a locally cyclic group , then is $\operatorname{Aut}(G)$ abelian?

Let $G$ be a locally cyclic group, then is it true that $\operatorname{Aut}(G)$ is abelian ? I know that $G$ has to be abelian but I cannot decide for $\operatorname{Aut}(G)$.
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### Listing elements of subgroup generated by $\{12,42\}$ in the integers with addition

The subgroup generated by these elements should contain both $12\mathbb{Z}$ and $42\mathbb{Z}$ but also ideals of the form $$(12k+42j)\mathbb{Z},\;j,k\in\mathbb{Z}$$ Is this the best answer I can ...
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### Centralizers of Elements in the Free Group

Let $F_n$ be the nonabelian free group on $n$ generators. According to what I have been reading from various sources online, the centralizer of some element $h \in F_n$, denoted as $C_{F_n}(h)$, is an ...
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### Isomorphism of invariant factor decomposition

By the structure theorem, for every finite abelian group $A$, we have an isomorphism $A \cong \mathbb{Z}_{d_1} \oplus \dots \oplus \mathbb{Z}_{d_n}$ for unique $d_i$, s.t. $d_i | d_{i+1}$. My question ...
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### Determine abelian groups with 48 elements

I am just doing some revision for my linear algebra exam, and I came across this problem: Determine all abelian groups (up to isomorphism) with exactly 48 elements. I am not sure I have ever ...
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### What is $pA$ when $p$ is a prime number and $A$ an abelian group?

Let $A$ be a finite abelian $p$-group. I want to prove that $pA$ is also an abelian finite $p$-group, of order strictly less than the order of $A$. The problem is that I don't even know what does the ...
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### Problem with the proof of abelian finite groups decomposition

I can't understand the proof which says that every finite abelian $p$-group can be written as a direct sum of cyclic $p$-groups. I'm using Lang's book of Algebra. My problem is about the following ...
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### Arnold's proof of Abel's theorem

I'm seeking help understanding this video. The author considers the equation $ax^5+bx^4+cx^3+dx^2+ex+f = 0$ and shows both the coefficients $a, b$... and solutions $x_1, x_2$... in the complex ...
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### kernel of product of group homomorphisms

Let $f,g:A \to B$ be group homomorphisms, with $B$ abelian. Then $f\cdot g$ is also a group homomorphism. What can I say about $\ker(f \cdot g)$ in terms of $\ker(f)$ and $\ker(g)$?
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### Simple characterization of integers among abelian groups

This is part of an early exercise in Freyd's abelian categories. Let $\mathscr{G}$ be the category of abelian groups. The group of integers is distinguished, up to isomorphism, by the facts that: ...
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### Are the groups $\mathbb R/ \mathbb Z$ and $\mathbb R^2 / (\mathbb Z \times \{0\} )$ isomorphic?

Is it true that as groups , $\mathbb R/ \mathbb Z \cong \mathbb R^2 / (\mathbb Z \times \{0\} )$ ? I only know that $\mathbb R \cong \mathbb R^2$ (as groups ) but I can see no way to decide whether ...
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### Non-abelian group with infinitely many abelian subgroups

I'm looking for a non-abelian group which has infinitely many abelian subgroups. Do you know any examples of such groups?
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### Is $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\cong 0$?

Since each 'generator' of $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}$ has the form $km\otimes_{\mathbb{Z}}\bar{a}=k\otimes_{\mathbb{Z}}m\bar{a}=k\otimes_{\mathbb{Z}}0=0$.
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### Elementary Abelian p groups

How to show that if any group of class 2 has only two conjugacy class sizes, then its centre and quotient by commutator both are elementary abelian?
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### Order of subgroup generated by two cyclic subgroups in $S_6$.

Let $S_6$ be the symmetric group, and $\alpha=(13456)$ and $\beta=(132)$ be its two permutations. How can we find the order of the subgroup generated by $\alpha$ and $\beta$. SOl: $\alpha^5$=...
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### Existence of open subgroup extending a smaller one

Let $G$ be an abelian topological group and $H \subseteq G$ a dense subgroup (equipped with the subset topology). Furthermore let $V \subseteq H$ be a subgroup that is open in $H$. Does there exist a ...
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### Quotient group of free abelian group

Let $A$ be a free abelian group, i.e. $A=\bigoplus_\alpha \mathbb Z$. Also let $B$ be a subgroup of $A$. Prove that $A/B\cong\mathbb Z$ implies $A=B\oplus \mathbb Z$. p.s. Actually this appears in ...
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### adding a relation to a group

We start with a group $G$ such that there is an abelian subgroup $\mathbb{Z}^3(e,f,g) \subset G$. Assume we have $G/\left\{ g = 1 \right\} \cong \mathbb{Z}^2(e,f)$. What can one say about $G$, except ...
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### Proving that the set of positive integers does not form a group under addition

The set of positive integers under addition has closure because it goes on for infinity and you will always have the element $a + b$. I am also aware that addition is associative but do we include $0$...
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### Are all complete groups abelian?

Hi: Let $G$ be a group and $G'$ it's commutator subgroup. Then $G > G' > 1$ is a series of normal subgroups of $G$. Suppose $G$ is complete. Then, if I'm not wrong, $Aut(G)$ is the stabilizer of ...
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### multiple of a crossed homomorphism from finite group to a divisible one is principal

Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed ...
Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category (i.e....