Page 114 - 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. 114
3.5 Nilpotent groups and p -groups
8. Fill in the details in Example 3.4.9.
9. Let N be a normal subgroup of a finite group G , and assume that |N | = n and
|G : N | = m are relatively prime. Let m1 be a divisor of m . Then a subgroup of G of
order m1 is contained in a subgroup of order m .
3.5 Nilpotent groups and p -groups
Nilpotent groups are groups which can be constructed from abelian groups by repeat-
edly forming central extensions. We exhibit some of the classical theory of these groups,
and show that they are closely related to finite p -groups. These form a very rich class
of groups. We prove that there are lots of finite p -groups, hence there is little hope to
classify them up to isomorphism.
3.5.1 Nilpotent groups
Definition and basic properties
We call 1 = G0 ⊂ G1 ⊂ · · · ⊂ Gn = G a normal series of G if each of its members is a
normal subgroup of G . A group G is nilpotent if it has a central series, i.e. a normal series
1 = G0 ⊂ G1 ⊂ · · · ⊂ Gn = G in which each factor Gi +1/Gi is contained in the center of
G /Gi . The length of the shortest central series of G is called the nilpotency class of G .
All nilpotent groups are solvable. Nilpotent groups of class no more than 1 are abelian.
The smallest solvable non-nilpotent group is S3.
Here is an example of how to manipulate nilpotent groups in GAP:
gap> l := AllSmallGroups(Size, 54, IsNilpotent, true);
[,
,
,
,
]
gap> NilpotencyClassOfGroup(l[2]);
1
gap> NilpotencyClassOfGroup(l[3]);
2
gap> ForAll(AllSmallGroups(54), IsNilpotent);
false
gap> G:= First(AllSmallGroups(54), x->not IsNilpotent(x));;
gap> StructureDescription(G);
"D54"
gap> List(l, StructureDescription);
[ "C54", "C18 x C3", "C2 x ((C3 x C3) : C3)", "C2 x (C9 : C3)",
"C6 x C3 x C3" ]
From the above example we observe that all nilpotent groups of order 54 can be
written as direct products of their Sylow p -subgroups. We will show later on that this
8. Fill in the details in Example 3.4.9.
9. Let N be a normal subgroup of a finite group G , and assume that |N | = n and
|G : N | = m are relatively prime. Let m1 be a divisor of m . Then a subgroup of G of
order m1 is contained in a subgroup of order m .
3.5 Nilpotent groups and p -groups
Nilpotent groups are groups which can be constructed from abelian groups by repeat-
edly forming central extensions. We exhibit some of the classical theory of these groups,
and show that they are closely related to finite p -groups. These form a very rich class
of groups. We prove that there are lots of finite p -groups, hence there is little hope to
classify them up to isomorphism.
3.5.1 Nilpotent groups
Definition and basic properties
We call 1 = G0 ⊂ G1 ⊂ · · · ⊂ Gn = G a normal series of G if each of its members is a
normal subgroup of G . A group G is nilpotent if it has a central series, i.e. a normal series
1 = G0 ⊂ G1 ⊂ · · · ⊂ Gn = G in which each factor Gi +1/Gi is contained in the center of
G /Gi . The length of the shortest central series of G is called the nilpotency class of G .
All nilpotent groups are solvable. Nilpotent groups of class no more than 1 are abelian.
The smallest solvable non-nilpotent group is S3.
Here is an example of how to manipulate nilpotent groups in GAP:
gap> l := AllSmallGroups(Size, 54, IsNilpotent, true);
[
gap> NilpotencyClassOfGroup(l[2]);
1
gap> NilpotencyClassOfGroup(l[3]);
2
gap> ForAll(AllSmallGroups(54), IsNilpotent);
false
gap> G:= First(AllSmallGroups(54), x->not IsNilpotent(x));;
gap> StructureDescription(G);
"D54"
gap> List(l, StructureDescription);
[ "C54", "C18 x C3", "C2 x ((C3 x C3) : C3)", "C2 x (C9 : C3)",
"C6 x C3 x C3" ]
From the above example we observe that all nilpotent groups of order 54 can be
written as direct products of their Sylow p -subgroups. We will show later on that this
