All Questions
Tagged with cayley-graphs finite-groups
10
questions
20
votes
4
answers
2k
views
Cayley graph of $A_5$ with generators $(1,2,3,4,5),(1,4,3,2,5)$
The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)?
The best I could get by trial and error is an embedding ...
- 24.5k
11
votes
4
answers
1k
views
Is there a Cayley graph of a non-abelian finite group that is not isomorphic to any Cayley graph of any abelian group?
It's the first question I post here :) I'm sorry if the question is too specific or if it's somehow repeating others.
In other words, my question is the following. Consider a Cayley graph $\Gamma$ of ...
- 111
6
votes
1
answer
244
views
Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite?
Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $\lvert\cdot\rvert$ the word length with respect to this set $S$.
Question. Is the function $k(g,h) = \...
- 2,126
5
votes
1
answer
312
views
$C_4\times C_2 : C_2$: what does this mean?
I am reading this paper where the object $C_4\times C_2 : C_2$ is used as a group structure. I know that $C_n$ is a cyclic group but don't know what kind of operation between groups is identified by ...
- 219
4
votes
2
answers
416
views
Transposition Cayley graphs are planar
Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...
- 1,841
4
votes
1
answer
249
views
Diameter of Cayley graphs of finite simple groups
Babai, Kantor and Lubotzky proved in 1989 the following theorem (Sciencedirect link to article).
THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ ...
- 225
4
votes
1
answer
144
views
Diameter for permutations of bounded support
Let $S\subset \textrm{Sym}(n)$ be a set of permutations each of which is of bounded support, that is, each $\sigma\in S$ moves $O(1)$ elements of $\{1,2,\dotsc,n\}$. Let $\Gamma$ be the graph whose ...
- 19.1k
3
votes
0
answers
211
views
Growth functions of finite group - computation, typical behaviour, surveys?
Looking on the growth function for Rubik's group and symmetric group, one sees rather different behaviour:
Rubik's growth in LOG scale (see MO322877):
S_n n=9 growth and nice fit by normal ...
- 22.9k
1
vote
0
answers
83
views
Example of family of Cayley graphs with Ramanujan behaviour on finite $p$-groups
This is a very general question: are there known examples of Ramanujan behaviour of Cayley graphs obtained from family of finite p-groups?
${\mathrm{\bf Adjacency~matrix:}}$ Given a graph ${\mathcal{G}...
- 405
-1
votes
1
answer
195
views
Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques
Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
- 1,841