Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-II
49
Unit-II
Definition:
Composition Series
A series of subgroups G = G 0
∆ G 1
∆ G 2
∆ ---- ∆ G r
= (1)
of a group G is called a Composition series of G if
and
(1) G i + 1
G i
for every i
(2) if each successive quotient G i
/ G i + 1
is simple
The above composition series is said to have length r. The successive quotients of a composition series are
called the Composition factors of the series.
Examples:
≅ Z2
1. Consider the symmetric group S 5
. It has a normal subgroup A 5
which is simple from unit I. Since S 5
A5
is also simple, we see that A 5
S 5
∆ A 5
∆ (1) is a composition series of S 5
. This is the only composition series
of S 5
, because only non-trivial proper normal subgroup of S 5
is A 5
.
2. Consider S 4
, From unit I we have
S 4
∆A 4
∆V 4
∆E 4
∆(1), the composition peries of S 4
.
H 1
∆ H 2
∆H 3 Recall ∆ − −V− 4
= − {(1), ∆H r (12) = ( 1(34), ) (13) (24), (14) (23)}
is Klein’s four group and
E 4
= {(1), (12) (34)}, Further
S4
A
4
A4
V4
E4
= 2 , = 3 2 tells
V
, = = 2
E
, ( 1 )
4
4
each successive quotients S 4
A4
V4
, , and E 4
is of prime order, hence are simple.
A V E ( 1) Theorem 1:
Every finite group has a composition series.
Proof:
4
4
4
Let G be a finite group. Use induction on G . If G is a simple then G ∆( 1 ) is a composition series of G. So
let G be not simple, Hence G has some maximal normal subgroup H, which has a composition series
by induction. Since G H 1
is simple, so
G = G0 ∆H1 ∆H2 ∆ − − − − ∆ H r = ( 1) is a composition series of G.
Note that infinite groups need not have composition series. We can consider infinite cyclic group Z.
As every non-trivial sub group of infinite cyclic group Z is isomorphic to Z; as Z is not simple, we see that Z
has no simple subgroups. So we can not construct composition series of Z.