# 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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### Questions about Sylow $p$-groups

Question 1 Is it true that there is only one Sylow $p$-group in an abelian group? Question 2 If there is only one Sylow $p$-group, then it is normal, true? Because if $H\leq G$ is a Sylow $p$-group ...
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### Abelian Group Element Orders [duplicate]

I want to show that if a finite abelian group has elements of order $m$ and $n$ then it will have an element of order $\text{lcm}(m,n)$. First I proved the lemma if $a$ has order $m$ and $b$ has ...
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### A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian?

Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$\forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$ Then how prove $G$ is an abelian ...
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### A criterion for a group to be abelian [duplicate]

I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let $m$ ...
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### A sufficient condition for profinite groups

I know that Edwin Hewitt and Kenneth A. Ross (1970) show: Any compact Hausdorff torsion group is profinite. But I don't have the book, the proof seems long and I need only the case of abelian groups ...
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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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### compact Lie group with non-compact Lie subgroup? [duplicate]

Can there be compact Lie groups with non-compact subgroups? I thought that was not possible until I thought of the torus with the irrational rotations. So if one identifies $U(1)\times U(1)$ with the ...
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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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### 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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### Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
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### Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Let $F$ be a free abelian group over a set $S$ with respect to the function $\varphi \colon S \to F$. Identify the set ...
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### Question about accessibility of category of free abelian groups.

I've read, that the accessibility of the category of all free abelian groups is independent on basic set theory (say ZFC). What is the reason for that? And how can I interpret this result? Does it ...
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### $G$ be an infinite abelian group such that every proper non-trivial subgroup of $G$ is infinite and cyclic ; then is $G$ cyclic?

Let $G$ be an infinite abelian group such that every proper non-trivial subgroup of $G$ is infinite cyclic ; then is $G$ cyclic ? ( The only characterization I know for infinite abelian groups to be ...
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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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### Least symmetric group having a certain Abelian group as subgroup

Given an Abelian group $G\simeq\bigoplus_{k}\mathbb Z_{p^{n_k}_{k}}$, where $p_1\leq p_2\leq ...$ are primes, how to calculate the least symmetric group $S_n$ having a subgroup isomorphic to $G$?
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### For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$ i.e. $[-1] \ne [2^k] , \forall k >0$ in $\mathbb Z_n$ ? Is there any ( at least sufficient ) ...
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### Subgroups of finite abelian groups.

For every subgroup $H$ of a finite abelian group $G,$ there exists a subgroup $N$ of $G$ such that $G/N \cong H.$ I need to prove this or give a counter example. I am aware of isomorphism theorems ...
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### Proof that a transitive permutation group (G, X) with G abelian, is sharply regular

As the title states, the question is the following: Let (G, X) be a transitive permutation group, where G is abelian. Show that (G, X) is "sharply regular". First of all I want to notice that in my ...
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### Properties of finite abelian group

Let $G$ be a finite abelian group of order $n$ . Then choose the correct statement. a) If d divides n, then there exist a subgroup of $G$ of order $d$ b) If d divides n, then there exist an ...
How can I determine all the subgroups of a commutative group, write the Hasse diagram, using Frobenius-Stickelberger Theorem and the isomorphism to $\mathbb{Z}_m$ of a cyclic group? In particular, for ...