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 ?
H . W.
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Theorem 2 – 4 .
Let ( H1 ,? ) and ( H2 ,? ) be subgroups of the group ( G ,? ) . (H1 UH2 ,?) is also subgroup if
and Only if H1? H2 or H2? H1 .
Proof .
Suppose that (H1 U H2 , ? ) is subgroup , we must prove that H1 ? H2 or H2? H1 .
If H1?H2 and H2? H1.
Then ? a ? H1 – H2 and b ? H2 – H1 .
If a? b ?H1 , then ,
b = a-1? (a ?b )?H1 ,
which is contradiction .
If a ? b ?H2 , then ,
a = ( a? b )? b-1? H2 ,
which is contradiction.
Hence H1? H2 or H2? H1.
Conversely , suppose that ,
H1 ?H2 or H2? H1 , we must prove that (H1 U H2 , ? ) is subgroup .
If H1? H2 , then H1U H2 = H2 ,
then ( H1 U H2 ,?) is subgroup .
If H2?H1 , then H1 U H2 = H1 ,
then ( H1 U H2 ,? ) is subgroup .
Example .
If (Z12 , +12 ) is a group ,
( H1 = { [0] , [3] ,[6] , [9] } , +12 ) and ( H2 = { [0] , [6] } , +12 ) are subgroups of (Z12 ,+12 )?
Then H1 U H2 = H1 .
But if H3 = { [o] , [4], [8] } .
Then ( H1U H3 , +12 ) is not subgroup ?