All Questions
Tagged with abelian-groups gr.group-theory
29 questions with no upvoted or accepted answers
29
votes
0
answers
877
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.
...
- 35.2k
14
votes
0
answers
518
views
Is $\mathbb{Z}^{\omega}$ ever the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$?
Recall that the covering number $cov(B)$ is the least cardinal $\kappa$ such that $\kappa$ meagre sets cover the real line. Andreas Blass and John Irwin http://www.math.lsa.umich.edu/~ablass/bb.pdf ...
- 2,111
10
votes
0
answers
428
views
Sets which are unions of translates of each other but aren't single translates
I'm a hobbyist mathematician so any question I ask here might be at risk of closure. I hope this one is good enough, but I'm not sure. This is a continuation of two questions I asked on math....
- 145
8
votes
0
answers
309
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A $\mathsf{ZF}$ example of a nonreflexive group which is isomorphic to its double dual?
Given a group $G$ denote by $G^\ast=\mathrm{Hom}(G,\Bbb Z)$ its dual and by $j\colon G\to G^{\ast\ast}$ the canonical homomorphism $g\mapsto (f\mapsto f(g))$. A group is reflexive iff $j$ is an ...
- 3,011
8
votes
0
answers
1k
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Examples of uncountable abelian $p$-groups
Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable?
By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, and totally ...
5
votes
0
answers
444
views
Subgroups and quotients of an abelian pro-finite group
It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$.
I'm wondering whether there is a counterpart for profinite groups.
For example is it true ...
- 93
4
votes
0
answers
63
views
Possible questions about the Tate-Shafarevich subgroup of a Galois hypercohomology group?
$\newcommand{\wt}{\widetilde}$
Let $n=1,2$. There are infinite torsion abelian groups $H^1$, $H^2$ killed by some natural number $m$.
There are finite subgroups
$$ {\rm Sha}^1 \subset H^1,\quad ...
- 14.2k
4
votes
0
answers
113
views
Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 14.2k
3
votes
0
answers
70
views
Admissibility of Ulm's invariants
Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define
$$G_{\alpha}=pG_{\beta}.$$
If $\alpha$ is a limit ...
- 31
3
votes
0
answers
107
views
(Non)complete abelian groups in the “transfinite p-adic topology”
For an abelian group $A,$ a prime $p$ and an ordinal $\alpha,$ we recursively define $p^\alpha A$ as a subgroup of $A$ such that $p^0A=A,$
$$p^{\alpha+1}A=p(p^\alpha A) \hspace{5mm} \text{and} \...
- 1,484
3
votes
0
answers
251
views
Commuting real elements in finite groups
Let $p$, $q$, $r$ be three distinct odd primes, and $G$ a finite group with $|G|$ divisible by $p$, $q$, $r$ to the first power only. Let $x,y,z \in G$ be of order $p,q,r$ respectively. Assume
(a.) $[...
- 191
3
votes
0
answers
138
views
The group of sequences in $G^{\mathbb{N}}$ that converge to $(0,0,0,\dots)$
Let $G$ be a discrete abelian group and $G^{\mathbb{N}}$ be the direct product (with the product topology), which consists of sequences $(a_1,a_2,a_3,\dots)$ of elements of $G$.
Let $G^{\infty}$ ...
- 517
3
votes
0
answers
98
views
Hales' generalization of the stacked bases theorem (seeking a proof)
In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ...
- 2,992
2
votes
0
answers
193
views
Decomposition of finite abelian groups of even order if there is an involution
Let $G$ be a finite abelian group and $\sigma :G\rightarrow G$ an automorphism of order two ($\sigma\circ \sigma =id_G$). Denote by $F$ and $A$ the subgroups of fixed and anti-fixed points of $\sigma$ ...
- 813
2
votes
0
answers
408
views
Conceptual proof of fundamental theorem of finite abelian groups
I'm looking for a conceptual proof of the following statement:
Lemma: Let $G$ be a finite abelian $p$-group. Let $a$ be an element of maximal order. Then $G=\langle a \rangle \times H$ for some ...
- 2,751
2
votes
0
answers
177
views
Existence of fully supported element in a finite-dimensional vector space over $\mathbb{F}_p$ (and in finite abelian groups)
Let $V$ be an $n$-dimensional vector space over $\mathbb{F} = \mathbb{Z} / (p)$, the field of $p$ elements, $p$ a prime, with $\{v_1, \dotsc, v_n \}$ a basis for $V$. An element $x \in V$ is called &...
- 191
2
votes
0
answers
70
views
Alternating $n$-homomorphism on abelian group is skew of $n$-cocycle
Let $A$ be a finitely generated abelian group. Let $c$ be a 2-cocycle on $A$, where $A$ acts trivially on $\mathbb{C}^\times$. It is well-known that the skew-map
$$ c(a_1,a_2) \longmapsto \frac{c(a_1,...
- 1,813
2
votes
0
answers
176
views
Trivial Tate modules
Let $A$ be an abelian group, and $p$ a prime.
I'll call $$T_p(A) := \text{Hom}_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p}, A).$$
If $A$ is finite, then $T_p(A)$ is trivial, but the converse is not ...
user119470
2
votes
0
answers
96
views
A kind of cancellation ; exchange problem for groups
For which $(m,n,k,l) \in (\mathbb N\cup \{0\})^4$ , with $m\le n ; k\le l$ , does there exist a group $G$ with a finite subnormal series with torsion-free Abelian quotients such that $G \times \mathbb ...
user111524
2
votes
0
answers
100
views
Alternating bihomomorphism is skew of 2-cocycle - relative situation
Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ (i.e. $\Omega(g,g)=1$) arises as the skew $\kappa/\kappa^T$ of a 2-...
- 1,813
2
votes
0
answers
130
views
Existence of a transfinite sequence of abelian groups having a strange property
I am studying a paper which uses the following lemma. The context is irrelevant, as the lemma is only used as a technical trick and has no pointer to a reference or hint in the proof but its link to ...
- 9,111
1
vote
0
answers
134
views
Isomorphic quotients of a countably infinitely-generated free abelian group
Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...
- 11
1
vote
0
answers
125
views
Is the commutator of the holomorph of generalized quaternion group abelian?
Let $Q_{2^{n}} = \langle x, y \mathrel\vert x^{2^{n-1}}=y^4 = 1, x^{2^{n-2}}=y^2, y^{-1}xy = x^{-1} \rangle$ be the generalized quaternion group of order $2^{n}$.
Let $\operatorname{Hol}(Q_{2^{n+1}})$ ...
- 19
1
vote
0
answers
97
views
A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori
$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...
- 14.2k
1
vote
0
answers
74
views
Pure (ordered) subgroups
Let $H,G$ be abelian groups with $H \leq G$. We say that $H$ is a pure subgroup of $G$ if for every $n \in \mathbb N$ and $h \in H$ the following holds: If $h$ is $n$-divisible in $G$, then $h$ is $n$-...
- 111
1
vote
0
answers
109
views
Symmetric analogue of "alternating bihomomorphism is skew of 2-cocycle" theorem
Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ arises as the skew $\kappa/\kappa^T$ of a 2-cocycle $\kappa \in Z^2(G,...
- 1,813
1
vote
0
answers
242
views
How many subgroups of order $\prod_{1}^{n} p_{i}^{n_{i}}$ are there in the finite product of cyclic groups?
All of the following ${p_{i},q_{i}}$are prime numbers, ${n,m,k}$ are pre-assigned integers.
Consider the product of cyclic groups $\prod_{1}^{n} \mathbb Z_{p_{i}^{n_{i}}}$ then we asked the question:
...
- 8,071
0
votes
0
answers
90
views
Invariants of primary groups
In Kaplansky's book "Infinite Abelian Groups", an abelian group $G$ is called primary if every element has order power of $p$ for some fixed prime number $p$. It is well-known that every ...
- 31
0
votes
0
answers
197
views
Quotients of Abelian Groups
Let $G$ be an abelian group and let $A$ and $B$ be subgroups of $G$. Furthermore, let $C$ be a subgroup of $A \cap B$. I would like to find another subgroup $A+B \subseteq D \subseteq G$ so that $D/(...
- 205