Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

54
ADVANCED ABSTRACT ALGEBRA
k
e j ( ) ( )
k
G G N N
⊂ N
4. The centre Z (G) and the quotient group G Z ( G) are finite p – groups of strictly smaller order. So by
induction and using above parts of this thorem, we get G is solvable.
5. 1 ×H ≅ H is a solvable normal subgroup of G ×H, and
is also solvable. Hence from part
(3) G is solvable.
Nilpotent Groups
Definition:
Central Series of a group G:
A normal series G = G ∆ G ∆ − − − − ∆ G r
= ( ) of a group G is called a central series of G if, for
0 1
1
each i, G G i +1
is contained in the center of G G i +1
i.e.
Gi
G
i + 1
≤ Z
F H G I K J ∀
G G
i + 1
i
A group G is said to be nilpotent if it has a central series.
Examples: 1.
1. An abelian group G has the central series G > ( 1 ) , and abelian groups are nilpotent.
2. S 4
, S 3
, the symmetric groups of degree 4 and 3 are solvable groups but they are not nilpotent.
Recall
hence S 4
and S 3
are solvable.
But center of S , i = 3, 4, i. e. Z ( S ) = ( 1).
G F ∴ ⊆
H G I Z
K J
i
G
i + 1
i
G G
i +
1
does not hold ∀ i,
i
are subnormal series in which each factor is abelian and
SG
4
1
where G = S 4
or S 3
.
Remarks:
1. The least number of factors in a central series in G is called nilpotency class (or just the class) of G.
2. The condition G G
F
Gi
HG
+ 1
G x G g = G ∀ x ∈G
i
⊆ Z
i + 1 i + 1 i + 1
i
F H G I K J
G G
i + 1 i + 1
, and ∀ g ∈G.
G
x ∈
i
G
i + 1
I K J
F for any x G G i
∈
i
⊆ Z
G H G I
,
K J
is equivalent to the Commutator condition that
i + 1 i + 1