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Solvable Groups and Chains of Subgroups

A) Prove that if H is nontrivial normal subgroup of the solvable group G then there is a nontrivial subgroup A of H with A normal subgroup of G and A abelian.
b)Prove that if there exists a chain of subgroups G1<=G2<=.....<=G such that
G=union(from i=1 to infinity)of Gi and each Gi is simple, then G is simple

Part a of this problem is taken from Dumitt and Foote pg 106 no 11 and that's exactly how it is written..

Solution Preview

1. This statement is definitely false. Here is an example.
G=Z_15={0,1,2,...,14}, N={0,3,6,9,12}. N is a normal
subgroup of G, But |N|=5 is prime, so N does not have nontrivial

2. Proof:
Suppose G is not simple, then G contains a nontrivial normal subgroup N,
such that N is not equal to {e} and G.
For each Gi, we check ...

Solution Summary

Solvable Groups and Chains of Subgroups are investigated.