# Tagged Questions

**3**

votes

**1**answer

510 views

### Cyclic subgroups of finite abelian groups

I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one.
Are there simple formulas if one ...

**2**

votes

**1**answer

535 views

### Minimal generation for finite abelian groups

Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:
1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$,
2) With orders that are ...

**4**

votes

**1**answer

765 views

### For what finite groups is the cardinality of a minimal generating set well defined?

Recently I learned that the cardinality of a minimal set of generators of a finite $p$-group
$G$ is well defined namely it is equal to the dimension of $H^1(G,\mathbb{F}_p)$. Moreover, if
...

**5**

votes

**2**answers

602 views

### infinite group that maps onto dihedral group

The group is generated by $y_i$, $i=0, ...,p-1$
with relations
$y_0y_1=y_1y_2=...=y_{p-1}y_0$
$y_0y_2=y_1y_3=...=y_{p-1}y_1$
$\vdots$
$y_0y_{p-1}=y_1y_0=...y_{p-1}y_{p-2}$
I have run into this ...

**3**

votes

**1**answer

385 views

### Looking for deterministic criteria to generate the symmetric group?

So let $S_N$ be the symmetric group of degree $N$. We think of it as a permutation group via its
natural action on the set $T=\{1,2,\ldots,N\}$.
Say that $H\leq S_N$ is a subgroup which acts ...

**17**

votes

**0**answers

420 views

### Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon.
Consider an inductive family of finite groups:
$$
G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i ...

**21**

votes

**3**answers

935 views

### In an inductive family of groups, does the probability that a particular word is satisfied converge?

We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g_1, g_2, \ldots , g_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More ...