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Results tagged with gr.group-theory
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user 37555
Questions about the branch of algebra that deals with groups.
84
votes
Feit-Thompson conjecture
It is not true anymore that a proof of this conjecture would lead to significant simplifications. Peterfalvi proved in 1984 a weaker version of this conjecture, which suffices to get rid of the chapte …
14
votes
1
answer
555
views
Which finite simple groups can be characterized by their action on a small set?
It is well known that a finite 4-times transitive permutation group is Matthieu, symmetric, or alternating. Another way of stating this is that the set
$$
\Omega = \{(x_1, x_2, x_3, x_4), 1\leq x_i\le …
12
votes
1
answer
503
views
Can a large transitive permutation group need many generators?
let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have $\frac{d(G)\log|G|} …
12
votes
About positive upper density
Suppose that $(S-n)\cap S$ has upper density 0 for all $n\leq N$. Then the density of the set of integers $x$, such that $|[xN, (x+1)N]\cap S|\geq 2$ is also 0, hence the upper density of $S$ is $\leq …
12
votes
1
answer
1k
views
How many generators does a direct product of alternating groups need?
P. Hall gave a formula for the number of generators of $G^n$ for any finite simple group $G$. One famous example is the fact that $A_5^{19}$ is 2-generated, but $A_5^{20}$ is not. The question of comp …
11
votes
Why are Fuchsian groups interesting?
Fuchsian groups are important for the theory of dessin d'enfants in arithmetic geometry. Wolfart wrote a survey here: http://www.math.uni-frankfurt.de/~wolfart/Artikel/abc.pdf .
One typical applicati …
10
votes
Applications of logic to group theory?
One area where logic really helped group theory is the theory of zetafunctions of torsionfree nilpotent groups. Define $\zeta_G(s)=\sum_{U\leq G} (G:U)^{-s}$, where summation runs over all finite inde …
10
votes
Accepted
On the Upper Density of $C_2$ in finite groups
Pyber showed that the number of groups of order $n$ is $\leq n^{\frac{2}{27}\nu(n)^3+C\nu(n)^{3/2}}$, where $\nu$ is the highest power of a prime dividing $n$ and $C$ is an absolute constant. On the o …
10
votes
Accepted
Generalizing the Notion of Nilpotent/Abelian/Cyclic Numbers
A nilpotent group is the direct product of its Sylow subgroups, hence the nilpotency class of a nilpotent group equals the maximum of the nilpotency classes of its Sylow subgroups. A group of order $p …
8
votes
Simplicity of alternating group $A_n$
I prefer the proof going via 3-cycles. It is probably the least elegant proof, but it is the one which you probably would have found when considering the problem without any prior knowledge. Also the …
7
votes
Accepted
Is there a left-orderable profinite group?
There is no such ordering, which is compatible with the profinite topology in the following way: If $x<y$, then there are small neighbourhoods $U, V$ of $x, y$, such that $u<v$ for all $u\in U, v\in V …
7
votes
Consequences of the Inverse Galois Problem
I once saw an application of a solved case of the inverse Galois problem.
It is well known, that the Dedekind $\zeta$-function of a number field does not determine the number field up to isomorphy. I …
6
votes
0
answers
180
views
Explicit descriptions of self-replicating pro-$p$ groups
A group $G$ is called self-replicating, if there exists a finite index subgroup $H$, such that $H\cong G\times\dots\times G$. Maybe the most famous example of a self-replicating group is a subgroup $K …
6
votes
expressing permutations in terms of generators
In general the problem is very difficult. There has been quite some work on the diameter of the Cayleygraph of $S_n$, the best results being due to Helfgott-Seress for the general case, and Helfgott-S …
5
votes
Accepted
Is the affine group generically 2-generated?
In general not. Consider the homomorphism $\mathrm{Aff}(p)\rightarrow\mathbb{F}_p^*$ mapping $ax+b$ to $a$. The probability that two random elements generate a subgroup for which the induced map is st …