subgroup

SUBGROUPS gorse two
Definition 2 - 1 .
Let (G , ? ) be a group and H ? G be a nonempty subset of G . The
pair ( H , ? ) is said to be a SUBGROUP of (G, ? ) if ( H , ? ) is group .
Example .
( Z , + ) is subgroup of ( R , + ) .
Note .
Each group (G , ? ) has at least two subgroups .
( { e } , ? ) and ( G , ? ) are subgroups of ( G , ? ) , these two subgroups
are called trivial subgroups .
Exercise .
1) Find all subgroups of the group ( Z8 , +8 ) .
2) Find all subgroups of (S3, o) and ( S4 ,o).

Theorem 2-1.
Let ( G , ? ) be a group and ?? H? G . Then ( H , ? ) is a
subgroup of ( G , ? ) if and only if a , b ? H implies a ? b-1? H.
Proof .
If ( H ,? ) is a subgroup and a , b ? H ,
then b-1 ?H ,
So a ? b-1?H ?
Conversely ,

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H ??
Since a ? b-1? H whenever a , b ? H ,
So we can take a = b .
Then b ? b-1 = e ?H .
b-1= e? b-1? H for every b in H .
a? b = a ? (b-1)-1 ? H?
hence H is closed .
If a, b , c ?H , then a , b ,c ?G.
a? ( b ? c ) = ( a ? b ) ? c ?
? is associative on H .
Hence (H ,? ) is subgroup .
Theorem 2 – 2 .
The intersection of two subgroups of the group is subgroup .
Proof .
Suppose that ( H , ? ) and ( K , ? ) are subgroups of group ( G , ? ).
We must prove that ( H? K ,? ) is subgroup .
Since (H , ?) and ( K , ? ) are subgroups ,
Then ? e ? H and e? K
Hence e? H? K .
So H?K ?? .
If a , b ?H? K, then a , b ? H and a , b ? K ,
Hence a ?b-1 ?H and a ? b -1?K .
So a ? b-1? H ?K .
Implies ( H? K , ? ) is subgroup .
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Definition 2 – 2 .
Let ( G , ? ) be group , ( G , ? ) is called commutative group if and
only if ,
a? b = b ?a for all a , b ? G .
Example 1.
(R , + ) is commutative group , for
a+ b = b + a for all a , b ?R .
Example 2.
( S3 , o ) is not commutative group .
Definition 2 – 3 .
The center of a group ( G , ? ) , denoted by cent G , is the set
cent G = { c ?G | c? x = x ? c for all x ? G }.
Example 1.
Find center of the group ( Z , + ) .
If n ?Z ,
Then n + m = m + n for all m ? Z .
So cent Z = Z
Example 2.
cent S3 = { e } .
Exercise .
Find cent Z8 .
Note .
For any group ( G , ? ) , cent G ?? .

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Theorem 2 – 3 .




( cent G , ? ) is subgroup of each group ( G , ? ) .

Proof .
cent G ?? .
If a , b ? cent G , then for every x ? G ,
a? x = x ? a and
b? x = x ? b .
( a? b-1 ) ? x = a ? ( b-1? x )
= a ?( x-1 ? b )-1
= a ?( b? x -1)-1
= a ?( x? b-1 )
= ( a? x ) ? b-1
= ( x ? a ) ? b-1
= x * ( a * b-1)
Hence a ? b-1? cent G .
So ( cent G , ? ) is subgroup.
Exercise .
Is the union of two subgroups group ?