CHAINS gorse

CHAINS
After we study subgroups , we will study some consepts by joining
the subgroups .
Defintion 1-1 .
Let (G, *) be group , any finite sequence of subsets of G ,
{e } = H0? H1 ?H2? … ? Hn-1 ? Hn = G
is called CHAIN where all (Hi ,*) are subgroups of the group G .The integer n . is called the LENGTH of the chain .
Example (1).
{0} ? {(32)} ?{ (16)} ? { (8)} ? {(4)} ? {(2)} ? Z
{0} ? {(32)} ? {(8)} ? {(4)} ? {(2)} ? Z
{0} ? {(9)} ? {(3)} ? Z
{0} ? Z
are chains of the group ( Z , + ) .

Example (2) .
{[0]} ? Z6
{[0]} ? {{ 0] , [ 3] } ? Z6
{ [0]} ? { [o] ,[2] , [4] } ? Z6
are chains of the group (Z6 , +6 ) .
Exercise (1).
Find all chains of Z12 , S3 and the group of symmetries of the sequare.
1

Now we will study some kinds of chains .
Definition 1 – 2 .
A chain {e } = H0? H1 ?H2? … ? Hn-1 ? Hn = G
is called NORMAL CHAIN if each group (Hi , *) is normal subgroup of (Hi+1 ,*) . o ?i ? n.
Exercise (2).
Which of the chains in examples (1) and (2) is normal chain ?
Exercise (3).
Find all normal chains in exercise (1).
Definition 1-3 .
A chain {e} = K0? K1? … ? Km-1? Km = G
is said to be a REFINEMENT or LENGTHMENT of the chain

{e } = H0? H1 ?H2? … ? Hn-1 ? Hn = G
provided there exists a one - to –one function f from { 0 , 1 ,2 , … , n }
into { 0 , 1 , 2 , … , m} such that Hi = Kf(i) for all i .
Notes
1) Every Hi conside with one of the Ki .
2)n? m .
Definition 1 – 4 .
In the group (G, * ) , the chain
{e } = H0? H1 ?H2? … ? Hn-1 ? Hn =G
forms a COMPOITION CHAIN for ( G , * ) provided
1)Hi is a subgroup of G , 2) Hi is normal subgroup of Hi+1 ,
3)If Hi? K ? Hi+1 , where K is normal subgroup of Hi+1 , then either K = Hi or
K = Hi+1. 2
Example .
The chains
{[0]} ? {[ 0] , [ 3] } ? Z6
{ [0]} ? { [o] ,[2] , [4] } ? Z6
are composition chains of ( Z6, +6 ) , but
{[0]} ? Z6 is not .
Example .
We can draw diagram of composition chains of ( Z30 , +30 ) as :

{[0]} ? <[ 6 ]> ? <[ 2]> ? Z30
{[0]} ? <[ 6 ]> ? <[ 3]> ? Z30
{[0]} ? <[ 10]> ? <[ 2]> ? Z30
{[0]} ? <[ 10]> ? <[ 5]> ? Z30
{[0]} ? <[ 15]> ? <[ 3]> ? Z30
{[0]} ? <[ 15]> ? <[ 5]> ? Z30
3
Exercise .
Find all composition chains of the groups in exercise (1) .
Definition 1 – 5 .
A group (G ,*) is SIMPLE GROUP if and only if it has no normal . subgroups other than the two trivial ones .
Example .
( Z3 , +3 ) is simple group .
Definition 1 – 6 .
A normal subgroup (H , *) is called a MAXIMAL NORMAL SUBGROUP . of the group ( G , *) if H ? G and ?normal subgroup K such that
H? K ? G .

Example (1) .
Ze is maximal of ( Z , + ) .
Example (2) .
{ [0] , [2] , [4] }is maximal of Z6 ,
also { [0] , [3] } is maximal of Z6 .
Notes .
1) The group may be have many distance maximal normal subgroups . . 2) A chain
{e } = H0? H1 ?H2? … ? Hn-1 ? Hn =G
is composition chain o f G if each subgroup (Hi , *) is a maximal . normal subgroup of ( H I+1 , *) .
Example .
{ [0]} ? { [o] ,[2] , [4] } ? Z6
{ [0] } is maximal of { [o] , [2] , [4] ], and
{ [0] , [2] , [4] } is maximal of Z6 .
Theorem 1 - 1
A normal subgroup ( H , *) of the group (G , *) is maximal if and only . if ( G / H , ? ) is simple . 4
Proof .
For each normal subgroup (K ,*) of G ,with H? K,
? corresponds a normal subgroup of G/H ?
This correspondence is 1 – 1 .
Since H is maximal ,
Then K = H or K = G .
Hence H is maximal iff ? exactly two normal subgroups in G / H
{ H and G / H } .
So H is maximal iff G /H is simple