Page 73 - 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. 73
mož Moravec: Some Topics in the Theory of Finite Groups 61
is concatenation, followed by reduction of terms of the form x ±1x ∓1 if necessary. For
further details we refer to [8].
Let X be a set and let F be a free group on X . Choose a subset Y of F , and let R = 〈〈Y 〉〉
be its normal closure in F . Then we say that the group G = F /R is given by generators X
and relations Y . We write G = 〈X | Y 〉.
The following result is simple but useful in recognizing groups from their presenta-
tions:
Lemma 3.2.6 (von Dyck’s Lemma) Let G be a group generated by x1, . . . , xm satisfying
relators r1 = 1, . . . , rn = 1. Let H be a group generated by y1, . . . , ym , and suppose that
ri (y1, . . . , ym ) = 1 for all i = 1, . . . , n . Then there exists a uniquely determined epimorphism
φ:G →H with x φ = yj for all j = 1, . . . , m .
j
A sample application von Dyck’s lemma will be given in the next section.
3.2.2 Examples of groups and GAP
In this section we present some important examples of groups. Along the way we show
how to use GAP to construct groups and study their properties. More information on how
to obtain GAP and apply its commands can be found at [5].
Cyclic groups
A group generated by one element is called a cyclic group. If G is a cyclic group, two
possibilites can occur. Either G is infinite, in which case it is isomorphic to ( , +), or it
is finite of order n , in which case it is isomorphic to ( n , +). In multiplicative notation,
cyclic groups will be denoted by C∞ and Cn , respectively.
In general, if G is an arbitrary group and g ∈ G , then the order of the cyclic subgroup
〈g 〉 of G is called the order of g , and denoted by |g |.
In GAP, one can construct cyclic groups in several different ways. The standard one
is as follows:
gap> G := CyclicGroup( 6 );
gap> Elements( G );
[ of ..., f1, f2, f1*f2, f2^2, f1*f2^2 ]
The list of the elements above may be a bit unexpected, as it does not indicate that
the group in question is cyclic. Rather, it reflects the fact that C6 is isomorphic to C2 ×C3,
and f1 and f2 are the corresponding generators of these factors.
It is possible to examine basic properties of the group we constructed above:
is concatenation, followed by reduction of terms of the form x ±1x ∓1 if necessary. For
further details we refer to [8].
Let X be a set and let F be a free group on X . Choose a subset Y of F , and let R = 〈〈Y 〉〉
be its normal closure in F . Then we say that the group G = F /R is given by generators X
and relations Y . We write G = 〈X | Y 〉.
The following result is simple but useful in recognizing groups from their presenta-
tions:
Lemma 3.2.6 (von Dyck’s Lemma) Let G be a group generated by x1, . . . , xm satisfying
relators r1 = 1, . . . , rn = 1. Let H be a group generated by y1, . . . , ym , and suppose that
ri (y1, . . . , ym ) = 1 for all i = 1, . . . , n . Then there exists a uniquely determined epimorphism
φ:G →H with x φ = yj for all j = 1, . . . , m .
j
A sample application von Dyck’s lemma will be given in the next section.
3.2.2 Examples of groups and GAP
In this section we present some important examples of groups. Along the way we show
how to use GAP to construct groups and study their properties. More information on how
to obtain GAP and apply its commands can be found at [5].
Cyclic groups
A group generated by one element is called a cyclic group. If G is a cyclic group, two
possibilites can occur. Either G is infinite, in which case it is isomorphic to ( , +), or it
is finite of order n , in which case it is isomorphic to ( n , +). In multiplicative notation,
cyclic groups will be denoted by C∞ and Cn , respectively.
In general, if G is an arbitrary group and g ∈ G , then the order of the cyclic subgroup
〈g 〉 of G is called the order of g , and denoted by |g |.
In GAP, one can construct cyclic groups in several different ways. The standard one
is as follows:
gap> G := CyclicGroup( 6 );
gap> Elements( G );
[
The list of the elements above may be a bit unexpected, as it does not indicate that
the group in question is cyclic. Rather, it reflects the fact that C6 is isomorphic to C2 ×C3,
and f1 and f2 are the corresponding generators of these factors.
It is possible to examine basic properties of the group we constructed above:
