# Tagged Questions

**10**

votes

**0**answers

239 views

### Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability ...

**7**

votes

**1**answer

387 views

### Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...

**25**

votes

**2**answers

881 views

### A group-theoretic perspective on Frankl's union closed problem

Here is a group theoretic phrasing of a special case of the union closed conjecture:
Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...

**2**

votes

**2**answers

294 views

### If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true ?

Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real"(symmetric) ? And vice verse ?
...

**7**

votes

**0**answers

219 views

### Why would dim primitive irrep divide size of some conjugacy class ?

From Isaacs et.al. 2005
Conjecture C. Let χ be a primitive
irreducible character of an arbitrary
ﬁnite group G. Then χ(1) divides |
clG(g)| for some element g ∈ G.
Here, of course, we ...

**18**

votes

**0**answers

675 views

### Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...

**3**

votes

**3**answers

735 views

### Bounds on number of conjugacy classes in terms of number of elements of a group ?

What are bounds on number of conjugacy classes in terms of number of elements of a group ?
(I allowed myself to edit the question in spirit of remarkable answers given to it by Gerry Myerson and ...

**8**

votes

**1**answer

471 views

### Periodic Automorphism Towers

In Scott's classic textbook on Group Theory, he asks:
Suppose that $G$ is a finite group. Is the sequence of isomorphism types
of the groups $Aut^{(n)}(G)$ for $n \in \mathbb{N}$ eventually periodic?
...

**10**

votes

**4**answers

870 views

### Diameter of universal cover

Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric).
What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or ...

**10**

votes

**2**answers

514 views

### A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...