# 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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### Characterization of the First Order Theory of Ordered Abelian Groups via Quantifier Elimination

In the paper "Elimination of Quantifiers in Algebraic Structures" Macintyre, McKenna and van den Dries, proved that every field (ordered field) whose theory admits quantifier elimination in the ...
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### On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case. Let $M$ be a free abelian group and $N$ a ...
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### Solving particular type of system of equations in $\mathbb R/\mathbb Z$

I apologize in advance for the long post. You can freely skip to the last paragraph. I was motivated by this question given in 5th grade mathematics competition that I was solving with my advanced ...
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### Abelian groups whose automorphism group is a $p$ group

$\def\Aut{\operatorname{Aut}}$ Let $G$ be a finite abelian group such that $\Aut(G)$ is an $p$ group ,that is, $|\Aut(G)|=p^n$ . Then can we determine the cyclic decomposition of $G$ or at least the ...
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### $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
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### Shift operator on locally compact groups

Assume $f:G\rightarrow H$ is a measurable function between two locally compact abelian groups and let $T^h(f) = f\circ T^h$, where $T^h(x) = x-h$ (group operations in G and H are written additively). ...
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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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### Why do mathematicians study elementary abelian groups?

I took two algebra courses that I liked as an undergraduate mathematics major in college, but we never covered elementary abelian groups. I recently got interested in the properties of a group I ...
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### Automorphisms of Abelian groups

Let $A$ be a free Abelian group and $N$ a characteristic subgroup of $A$ such that $A/N$ is finite. I also know that $Aut(A/N)$ and $Aut(N)$ are both finite. I have to prove that $Aut(A)$ is finite. ...
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### Show that $G$ is Abelian if and only if $f: G\times G \to G$ is a homomorphism.

Let $G$ be a group. Let $H$=$G\times G$ be the direct product of $G$ with itself. Define $f: H\to G$ to be $f((g,h))=gh$ for any $(g,h)\in H$. Show that $G$ is Abelian if and only if $f$ is a ...
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### Structure theorems for infinitely generated Abelian groups

The classification theorem for finitely generated Abelian groups is well known and plays big role in mathematics. Are there any structure theorems about infinitely generated Abelian groups known?
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### What triple “tensor product” is this? Is it just isomorphic to a double tensor product?

Consider the abelian groups $A = \Bbb{Q}^{\times}, B = \Bbb{Q}^{\times}, C = \Bbb{Z}^+$. What if we formed a product like: $A \star B \star C = \text{Free}_{\Bbb{Z}}(A \times B \times C)$ ...
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### Isomorphic product of finite abelian groups

Suppose $X,Y,Z$ are finite abelian groups with $X \times Y \cong X \times Z$. How to show that $Y\cong Z$? If we assume that we can decompose $Y,Z$ into cyclic groups that are powers of primes, I ...
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### Find all abelian groups that fit in a given short exact sequence.

I have to find all abelian groups that can appear in this short exact sequence. $0\rightarrow \mathbb{Z} \rightarrow A \rightarrow \mathbb{Z}\oplus\mathbb{Z}_5 \rightarrow 0$ First of all since ...
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### Structure of $p$-torsion groups

Let $A$ be an abelian group whose elements have order a power of certain prime $p$, suppose the $p$-torsion elements are finite, must $A$ be of the form (\mathbb{Q}_p/\mathbb{Z}_p)^{\oplus r}\oplus ...
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### Isomorphism Type of $\mathbb{Z_8}\times\mathbb{Z_6}\times\mathbb{Z_4} /\langle (2,2,2) \rangle$

Determine the isomorphism type of $\mathbb{Z_8}\times\mathbb{Z_6}\times\mathbb{Z_4}/\langle (2,2,2) \rangle$. Give two proofs: one using elementary analysis of orders of elements, and the other using ...
### Finding a property for $G/Z(G)$ where $G$ is a nonabelian group
If $G$ is non-abelian group and $Z(G)$ is it's center, what is the least property for $G$ such that $\frac{G}{Z(G)}$ is abelian?
### If $X$ is an Abelian group, then $\ker_X : \mathrm{Cong}(X) \rightarrow \mathrm{Sub}(X)$ is a bijection. Is there a partial converse?
(All monoids are written additively in this question, even the non-commutative ones.) Given a monoid $X$, write $\mathrm{Sub}(X)$ for the lattice of submonoids of $X$, and write $\mathrm{Cong}(X)$ ...