Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-I
25
1
Note that αβ =
2
2
1
3
1
≠
3
3
2
2
3
1
= βα
Hence S
3
, the group of 6 elements, called symmetric group which is non-abelian. This is the smallest finite
non-abelian group, since groups of order 1, 2, 3, 5 are of prime order, hence cyclic and, therefore, they
are abelian. A group of order 4 is of two types upto isomophism, either cyclic or Klein 4-group, given in
example 2.
Cycle Notation
1
2 3 4
Let σ =
3
4 1 6
This can be seen as:
5
5
6
2
1
2
5
σ
σ
( { )( ( )}) ()( ( ) ( )( }{ ) )( ( )}) )( )}{ ( )}
σS H A 1 =
13 1 e,
2123
246
S,
13 , H
5132
2,
= 2313
e
,,
1123 13 246 2
, H132
3
3=
e,
23
2 1
β [
3
3
S=
3
: A3
] = ,
= αβ 2,
= ,
α β =
1
3 A 23
2
1 3
3
2
2
3
,
1
σ
6
4
In cycle notation σ can be written as
Therefore from example 6:
It has 4 proper subgroups:
and
so A 3
is a subgroup of S 3
of index 2. It can be easily verified that A3 ∆ S3
. Infact, it can
be generalised, that every subgroup of index 2 is a normal subgroup in its parent group.
group.
is called alternating