Jordan-Holder

Question 1

What is a simple group

Answer 1

A group with no proper normal subgroup

Question 2

what is a maximum normal subgroup.

Answer 2

A proper normal subgroup A is called a maximum normal subgroup of G if A◃H◃Gimplies H=G or H=A
Note A is a maximum invariant normal subgroup if and only if G/Ais a simple group, because H/A is a normal subgroup of G/A

Question 3

what is a composition series

Answer 3

IfGis not simple, letAa maximum normal subgroup inG. Now ifAis not simple, letA1be a maximum normal subgroup. Continuing in this fashion we can construct a sequence, called acomposition seriesas follows.

G▹A▹A1▹…▹Ar▹{1}G▹A▹A1▹…▹Ar▹{1}

whereG/A,A/A1,A1/A2,…,ArG/A,A/A1,A1/A2,…,Arare all simple nontrivial groups, which are called thecomposition quotient groups. The orders of the composition quotient groups are called thecomposition indices.

Question 4

what is the jordan holder theorem

Answer 4

In any two composition series for a groupGG, the composition quotient groups are isomorphic in pairs, though may occur in different orders in the sequences.

Question 5

what is a soluble group

Answer 5

A groupGis said to besolubleif all the composition indices ofGare prime

Question 6

composition indices

Answer 6

’’’’

Question 7

lemma 1

Answer 7

If a normal subgroupH ofAnforn≥3contains a cycle of degree3thenH=An’

Proof:Without loss of generality let(123)∈H. Forn=3,(123)generatesA3and there is nothing to prove. Forn>3, sinceH is normal, it must also contains^−(1)(123)sfor any even permutationss. Sets=(32k)fork>3. Then we have thatH contains(1k2), and hence also its square which is(12k).Recall these cycles generateAn.

Question 8

theorem 1

Answer 8

Anis simple forn>4.

Question 9

theorem

Answer 9

Corollary:Anis the only subgroup of order(1/2)n!inSnwhenn>4.