Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

52
ADVANCED ABSTRACT ALGEBRA
.
Hence we finally conclude that our two initial composition series of B are equivalent.
Definition:
A series of sub groups
G = G0 ∆ G1 ∆ G2 ∆ − − − − ∆ G s
= ( 1) of group G is called a subnormal series of G if Gi
+1
∆ Gi
for
each i.
A subnormal series is called a normal series of G if
Solvable groups:
First we define Commutators in a group G. Let a, b
∈ G. The element
Commutator and is denoted by [a, b]. The Commutator [a, b] = 1, only when ab = ba.
is called a
–
[ a, b] 1 = [ b, a],
i.e., the element, inverse to the Commutator is itself a Commutator. But a product of
Commutators need not be a Commutator. Thus, in general, the set of Commutators of a group is not a sub
group. The smallest sub group G 1
of the group G containing all Commutators is called its Commutator sub
group. Note that the commutator sub group G 1
is the set of all possible products of the form [a 1
, b 1
] ---- [a r
,
b r
], where a , b ∈ G , and r is a natural number. From
i
i
which, as a consequence, implies that G 1
∆ G.
Remarks:
1. The commutator sub group G ' of an abelian group is trivial.
2. The Commutator sub group of S n
is A n
, n ≥ 1.
3. The Commutator sub group of GL (n, F) is SL (n, F), F is a field.
4. The Commutator sub group A ' n of A n
is A n
, is A ' n = A n
, because the non-commutative group A n
,
has no non-trivial proper normal subgroups.
Theorem 4:
The Commutator sub group G ' of a group G is the smallest among the normal sub group H of the group G for
which is an abelian group.
b
C
n
G⇔
a
a,
a
Proof:
The Commutator [xH, yH] = [x, y] H
is trivial
is
is abelian
From [a, b] 2 = [a g , b g ], a, b, g ∈ G, we get the second Commutator sub group G", i.e. the Commutator sub