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Solvable Groups and Chains of Subgroups

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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..

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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
subgroups.

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.

$2.19