Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-II
57
we write the refinement
l0q∆ 72Z ∆8Z∆4Z ∆Z
(3)
of (1) and the refinement
l0q∆ 72Z ∆18Z∆9Z ∆Z
(4)
of (2)
Both refinements have four factor groups :
3. has
Z
4Z
4Z
8Z
≅ Z , ≅ Z , ≅ Z
8Z
72Z
4 2 9
72z
0
l q ≅
72Z or Z
4. has
Z
9Z
Z Z
Z 9 9 18
≅ 9 ≅ Z2 , ≅ Z4
,
18Z
72Z
mr
72Z
0
l q ≅
72Z or Z
Hence (3) and (4) have four factor groups isomorphic to Z 4
, Z 2
, Z 9
and 72Z or Z.
Z
Z Z 18Z
4Z 9Z
≅
4
≅ , ≅ Z2
≅ ,
4 72Z
8Z 18Z
Z8
Z Z 72 A
2
× Z
≅
2
Z
Z9
≅2 1 ≅ Z
,
2 ,
Z
≅2
× 1 1 × 1 ≅ Z
72Z
2
72Z
9Z
( 0) (or Z)
Note carefully the order in which the factor groups occur in (3) and (4) is different.
Exmple 5 :
mr mr mr
Consider G = V4 = Z2 × Z2
We write a norml series for G=V 4
:
G = Z2 × Z
2
∆ Z × 1 ∆ 1 × 1
2 mr mr mr
This is a composition series, because
But Z 2
is a simple group. Therefore, above normal series is a composition series. The composition factors for
G=V 4
= Z 2
× Z 2
are Z 2
and Z 2
Exmple 6 :
Let G=S 3
, a normal series for G is given by
G = S ∆ A ∆ 1
3 3
l q
l q
S A ≅ Z , A 1 ≅ Z
3 3 2 3 3
Both Z 2
and Z 3
are simple group. Therefore, normal series for S 3
is a composition series. The composition
factors for S 3
are Z 2
and Z 3
.