# Tagged Questions

**9**

votes

**1**answer

307 views

### Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups?

Let $X_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Also ...

**1**

vote

**0**answers

124 views

### Is a certain group, derivable from the surreal numbers, isomorphic to the surreal numbers?

Let's treat $\mathbf{No}$ as a group under addition, and forget its field structure for a little bit.
I will define a "maximally Archimedean subgroup" of $\mathbf{No}$ as a subgroup which is
...

**10**

votes

**3**answers

2k views

### A “mother of all groups”? What kind of structures have “mother of all”s?

For ordered fields, we have a “mother of all ordered fields”, the surreal numbers $\mathbf{No}$, a proper-class “field” which includes (an isomorphic copy of) every other ordered field as a subfield. ...

**12**

votes

**1**answer

383 views

### Without choice, can every homomorphism from a profinite group to a finite group be continuous?

In ZFC, some homomorphisms from profinite groups to finite groups are discontinuous. For instance, see the examples in this question. However, all three constructions given use consequences of the ...

**5**

votes

**1**answer

210 views

### Is the dual of the product of infinite cyclic groups a free abelian group ?

By a theorem of Specker, the group $Hom(\prod_{\aleph_0} \mathbb{Z},\mathbb{Z})$ is isomorphic to $\bigoplus_{\aleph_0}\mathbb{Z}$ and is in particular a free abelian group. I wonder, if this ...

**11**

votes

**1**answer

400 views

### counting non-isomorphic groups of a given cardinality

Given an infinite cardinal $\kappa,$ is there some nice way to construct $2^\kappa$ non-isomorphic groups of that cardinality? In the answer to this stackexchange question, there is a fairly ...

**4**

votes

**1**answer

551 views

### Can every nonempty set carry abelian group structure? [duplicate]

Possible Duplicate:
Does every non-empty set admit a group structure (in ZF)?
Let $X$ be an arbitrary nonempty set. Can you define a multiplication making it into an abelian group?
If ...

**3**

votes

**1**answer

325 views

### Universe-sized groups with only set-sized normal subgroups, their cardinality in a certain range

Let $\kappa$ be an inaccessible cardinal, and let $G$ be a group with $|G| \geq \kappa$. For any cardinal $\lambda \le \kappa$ (regular, say, but not necessary), say $G$ is $\lambda$-simple if for all ...

**0**

votes

**0**answers

188 views

### How many minimal pair-wise coverings can there be?

Suppose I have a set of finite sets: $X = \{V_1, V_2, V_3\}$ where each set called $V_i$ in $X$ contains a number of symbols (i.e. $V_1 = \{a,b,c\}$). $Z$ contains all of the Cartesian products of ...

**6**

votes

**2**answers

401 views

### Distinct, non-homeomorphic, profinite topologies on a given abstract group ?

Just a silly little question which arose in connection with infinite Galois groups and their Krull topology:- can a given abstract group be endowed with distinct, non-homeomorphic, profinite ...

**15**

votes

**1**answer

728 views

### What is the largest Laver table which has been computed?

Richard Laver proved that there is a unique binary operation $*$ on $\{1,\ldots,2^n\}$ which satisfies $$a*1 \equiv a+1 \mod 2^n$$
$$a* (b* c) = (a* b) * (a * c).$$
This is the $n$th Laver table ...

**1**

vote

**0**answers

216 views

### Modern books about orders and algebras on trees

Please help to find books about orders and algebras on trees.
If there is no modern books, please advice good old ones!
I'm more interested in finite trees (my current problem), but infinite ones are ...

**4**

votes

**1**answer

303 views

### When are unions of isomorphic groups isomorphic?

I was thinking about how to prove $\operatorname{Br}(K)\cong H^2(\operatorname{Gal}(\bar{K}/K),\bar{K}^*)$ without having to introduce inductive limits and all the profinite stuff. So, I started ...

**4**

votes

**2**answers

327 views

### Is every bounded representation of Z unitarisable when all sets are measurable?

For the purpuse of this question, a group is amenable iff there exists a Foelner sequence.
Dixmier unitarisability problem asks whether a (countable discrete) group G is amenable iff every bounded ...

**5**

votes

**1**answer

410 views

### Is it possible to construct (without choice, even?) a non-finitely-generated group with no proper non-finitely-generated subgroup?

Is there a non-finitely-generated group each of whose proper subgroups is finitely generated? If so, what form of choice (if any) is required to construct such a group?

**5**

votes

**3**answers

487 views

### covering groups by infinitely many cosets

The classical Neumann lemma states that if a group is covered by finitely many cosets, then at least one of these cosets is the coset of a subgroup of finite index. (Actually, the lemma says more, ...

**10**

votes

**3**answers

713 views

### Construction of a proper uncountable subgroup of $\mathbb{R}$ without Choice.

It is straightforward to construct proper uncountable subgroups of $\mathbb{R}$. One can construst a basis for $\mathbb{R}$ over $\mathbb{Q}$, and then there are many possibilities (just consider the ...

**6**

votes

**5**answers

1k views

### Naturally occuring groups with cardinality greater than the reals.

In group theory, the single most important piece of information about a group is its cardinality, which is of course either finite, countably infinite, or uncountably infinite. Usually, however, ...

**3**

votes

**1**answer

642 views

### Shelah's proof of the independence of the Whitehead Problem

In the beginning Shelah classifies all $\aleph_1$-free Abelian groups into 3 possibilities each of which is satisfied by some $\aleph_1$-free Abelian group and the classification depends on the group ...

**7**

votes

**8**answers

2k views

### Is there a ground between Set Theory and Group Theory/Algebra?

It is well known that there are strong links between Set Theory and Topology/Real Analysis.
For instance, the study of Suslin's Problem turns out to be a set theoretic problem, even though it started ...

**38**

votes

**5**answers

4k views

### Which graphs are Cayley graphs?

Every group presentation determines the corresponding Cayley graph, which has a node for each group element, and arrows labeled with the generators to get from one group element to another.
My main ...

**52**

votes

**3**answers

5k views

### Does every non-empty set admit a group structure (in ZF)?

It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...

**27**

votes

**4**answers

2k views

### Can the symmetric groups on sets of different cardinalities be isomorphic?

For any set X, let SX be the symmetric group on
X, the group of permutations of X.
My question is: Can there be two nonempty sets X and Y with
different cardinalities, but for which SX is
isomorphic ...

**2**

votes

**2**answers

369 views

### Automorphisms of the totally ordered group Z^n with lexicographical order

It is easy to see that the totally ordered group Z (the integers) with the natural order has no non-trivial automorphisms. Is this also true for Z^n with the lexicographical order?