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Results tagged with finite-groups
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user 9025
Questions on group theory which concern finite groups.
2
votes
Number of generators of a subgroup of a finite simple group
Every finite simple group can be generated by two elements. Except in the case of prime order, one of the elements can have order 2. See here for example.
- 37.7k
7
votes
1
answer
144
views
Covering a set with images of a transversal
Let $G$ be a permutation group on a finite set $\Omega$ with orbits $\Omega_1,\ldots,\Omega_k$.
By a transversal we mean a set $\lbrace\omega_1,\ldots,\omega_k\rbrace$ with $\omega_j\in\Omega_j$ for e …
- 37.7k
16
votes
Is there a Cayley graph of a non-abelian finite group that is not isomorphic to any Cayley g...
If I understand my own 1979 catalogue of small transitive graphs, this happens first at 12 vertices. The simplest example to describe (L10 in the catalogue): take the tetrahedon and cut off each of t …
- 37.7k
4
votes
Graph automorphism group
Peter and YCor already gave a counterexample, so this answer is just some additional commentary. I'll ignore loops for simplicity. If $\varGamma$ is a permutation group on $\lbrace 1,\ldots, n \rbrace …
- 37.7k
17
votes
0
answers
505
views
Maximum automorphism group for a 3-connected cubic graph
The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a g …
- 37.7k
15
votes
Which finite groups are not the automorphism group of some rooted finite tree?
If I remember correctly, the automorphism groups of trees are those groups which you can make from symmetric groups by direct products and wreath products. This is rather few groups.
An example of a …
- 37.7k