Page 89 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 89
mož Moravec: Some Topics in the Theory of Finite Groups 77
Examples of solvable groups include the following:
Theorem 3.2.36 Let p , q , r be primes. Then all groups of orders p m q n or pq r are solvable.
We skip the proof. Solvability of groups of order p m q n is also refered to as the Burnside’s
p m q n -theorem. It is proved using character theory.
A celebrated theorem by Feit and Thompson says that every group of odd order is
solvable. The proof is very long (about 255 pages) and represents a milestone in the
classification of finite simple groups as it was a first significant indication that such a
classification might be possible. We mention here that the Feit-Thompson theorem was
recently reproved using interactive theorem prover Coq.
3.2.7 How to draw a group?
In this section we assume the reader is familiar with basic terminology of graph theory.
Let G be a group generated by a set S. The Cayley graph Γ = Cay(G ,S) is a colored directed
graph given as follows: the vertex set of Γ is identified with G . To each s ∈ S we assign
a color cs . The vertices g and s g are joined by a directed edge of color cs for all g ∈ G
and s ∈ S. The set S is usually assumed to be finite, symmetric (i.e., S = S−1) and not
containing the identity element of the group. In this case, the uncolored Cayley graph
is an ordinary graph: its edges are not oriented and it does not contain loops (single-
element cycles).
One can modify the above definiton to the case when S is a set of elements of G
that does not generate G . We still get a graph, but it may not be connected. From the
definition of Cayley graphs it also follows that the Cayley graph of a given group clearly
depends on the choice of a generating set S. Here are some examples that illustrate this.
Example 3.2.37 If we take the cyclic group Cn = 〈x 〉 of order n and S = {x , x −1}, then
Cay(Cn ,S) is an undirected cycle n of length n . If we take S = {x }, then, unless n = 2, the
corresponding Cayley graph is a directed cycle of length n. In the case n = 5 the diagram is
as follows:
1
x4 x
x3 x2
Examples of solvable groups include the following:
Theorem 3.2.36 Let p , q , r be primes. Then all groups of orders p m q n or pq r are solvable.
We skip the proof. Solvability of groups of order p m q n is also refered to as the Burnside’s
p m q n -theorem. It is proved using character theory.
A celebrated theorem by Feit and Thompson says that every group of odd order is
solvable. The proof is very long (about 255 pages) and represents a milestone in the
classification of finite simple groups as it was a first significant indication that such a
classification might be possible. We mention here that the Feit-Thompson theorem was
recently reproved using interactive theorem prover Coq.
3.2.7 How to draw a group?
In this section we assume the reader is familiar with basic terminology of graph theory.
Let G be a group generated by a set S. The Cayley graph Γ = Cay(G ,S) is a colored directed
graph given as follows: the vertex set of Γ is identified with G . To each s ∈ S we assign
a color cs . The vertices g and s g are joined by a directed edge of color cs for all g ∈ G
and s ∈ S. The set S is usually assumed to be finite, symmetric (i.e., S = S−1) and not
containing the identity element of the group. In this case, the uncolored Cayley graph
is an ordinary graph: its edges are not oriented and it does not contain loops (single-
element cycles).
One can modify the above definiton to the case when S is a set of elements of G
that does not generate G . We still get a graph, but it may not be connected. From the
definition of Cayley graphs it also follows that the Cayley graph of a given group clearly
depends on the choice of a generating set S. Here are some examples that illustrate this.
Example 3.2.37 If we take the cyclic group Cn = 〈x 〉 of order n and S = {x , x −1}, then
Cay(Cn ,S) is an undirected cycle n of length n . If we take S = {x }, then, unless n = 2, the
corresponding Cayley graph is a directed cycle of length n. In the case n = 5 the diagram is
as follows:
1
x4 x
x3 x2
