# Questions tagged [abelian-groups]

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19
questions

**89**

votes

**2**answers

6k views

### $A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?

**11**

votes

**2**answers

2k views

### Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?

**33**

votes

**4**answers

8k views

### When is Aut(G) abelian

let $G$ be a group such that $\mathrm{Aut}(G)$ is abelian. is then $G$ abelian?
This is a sort of generalization of the well-known exercise, that $G$ is abelian when $\mathrm{Aut}(G)$ is cyclic, but ...

**26**

votes

**0**answers

712 views

### The field of fractions of the rational group algebra of a torsion free abelian group

Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions.
...

**14**

votes

**1**answer

893 views

### Is the class of additive groups of rings axiomatizable?

I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ...

**9**

votes

**2**answers

474 views

### Is there a non-degenerate quadratic form on every finite abelian group?

Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, ...

**12**

votes

**1**answer

876 views

### Classification of symtrivial modules over a PID

Let us call a module $M$ over a commutative ring $R$ symtrivial if the symmetry $M \otimes M \to M \otimes M, a \otimes b \mapsto b \otimes a$ equals the identity (the same notion applies to arbitrary ...

**19**

votes

**4**answers

1k views

### Constructively, is the unit of the “free abelian group” monad on sets injective?

Classically, we can explicitly construct the free Abelian group $\newcommand{\Z}{\mathbb{Z}}\Z[X]$ on a set $X$ as the set of finitely-supported functions $X \to \Z$, and so easily see that the unit ...

**9**

votes

**0**answers

278 views

### An abstract zero-sum problem

I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...

**5**

votes

**2**answers

580 views

### What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$?

What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$?
I have a reference that says the group in question is $\mathbb{Q}_p,$ the additive group of the quotient field of the $p$-adic integers. Can ...

**16**

votes

**1**answer

991 views

### A possible mistake in Walter Rudin, "Fourier analysis on groups"

I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$):
Suppose $E$ is a coset in $\Gamma_2$ ...

**6**

votes

**0**answers

244 views

### When is every element of a coend of abelian groups contained in one of the summands?

Let $I$ be a small category and let $D : I^{\mathrm{op}} \times I \to \mathsf{Ab}$ be a functor. The coend
$$\int^{i \in I} D(i,i)$$
can be constructed as the direct sum $\bigoplus_{i \in I} D(i,i)$ ...

**12**

votes

**1**answer

635 views

### How nearly abelian are nilpotent groups?

It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2
Can this sentiment be made precise
in the sense of the
Turán and Erdős definition of "the probability that two elements of ...

**5**

votes

**1**answer

289 views

### Quotient groups obtained by quotienting $G^n$ by $G^{n-1}$

Notation: For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself.
Problem set up:
Consider, for a finite group $G$, and $n > 1$, the set $Q(G)_n$ of all ...

**2**

votes

**1**answer

586 views

### One-dimension Algebraic groups

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. ...

**4**

votes

**2**answers

421 views

### Number of torsion-free abelian groups

Let $\mathfrak{c}$ be the cardinality of the continuum. How much Choice, if any, is needed to prove that there are $2^{\mathfrak{c}}$ distinct (mutually nonisomorphic) torsion-free abelian groups of ...

**4**

votes

**2**answers

393 views

### Non-torsion part of the abelianisation of congruence subgroups

I've posted this question on math.stackexchange, but haven't gotten any responses so I'm trying here instead.
Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite ...

**3**

votes

**1**answer

260 views

### Finite dimensional compact abelian group that is not a product of connected and a totally disconnected

Let $G$ be a compact abelian group. A compact abelian group is said to have dimension $n$ if $\dim_\mathbb{Q} \mathbb{Q}\otimes \hat G = n$. Equivalently one can show that this holds if $G$ is ...

**2**

votes

**2**answers

434 views

### Non-archimedean group over the reals

I have a totally ordered group $(\mathbb{R};\leq,\oplus,0,-)$ with the reals as base set satisfying monotonicity, i.e.
for all $x,y,z$ we have that if $y\leq z$ then $x\oplus y \leq x\oplus z$, and I ...