Proving a Subgroup of a Solvable Group is Solvable

[samkolb]

I'm supposed to show that a subgroup of a solvable group is solvable.

(I am using the Fraleigh Abstract Algebra book and the given definition of a solvable group is a group which has a COMPOSITION series in which each of the factor groups is abelian. In other books I have looked at a solvable group is defined as one which has a SUBNORMAL series in which all the factor groups are abelian.)

Let K be a subgroup of a solvable group G. Let

{e}=Ho<H1<...<Hn=G be a composition series for G in which all the factor groups are abelian.

I showed that the distinct groups from among (K & Hi) (&= set intersection)
form a subnormal series for K.

Then I used the second isomorphism to show that (K & Hi)/(K & Hi-1) is isomprphic to Hi-1(K & Hi)/Hi-1.

I then showed that Hi-1(K & Hi) is a subgroup of Hi. Since Hi/Hi-1 is abelian, Hi-1(K & Hi)/Hi-1 is abelian, and so (K & Hi)/(K & Hi-1) is abelain.

What I can't show is that the distinct groups among (K & Hi) from a composition series for K. That is, I can't show that (K & Hi)/(K & Hi-1) is simple.

Please help!

 

[matt grime]

(I am using the Fraleigh Abstract Algebra book and the given definition of a solvable group is a group which has a COMPOSITION series in which each of the factor groups is abelian. In other books I have looked at a solvable group is defined as one which has a SUBNORMAL series in which all the factor groups are abelian.)

These are the same: factor groups only exist for normal subgroups.

Let K be a subgroup of a solvable group G. Let

{e}=Ho<H1<...<Hn=G be a composition series for G in which all the factor groups are abelian.

I showed that the distinct groups from among (K & Hi) (&= set intersection)
form a subnormal series for K.

Then I used the second isomorphism to show that (K & Hi)/(K & Hi-1) is isomprphic to Hi-1(K & Hi)/Hi-1.

please could you tex this clearly so people can understand what you mean?

What I can't show is that the distinct groups among (K & Hi) from a composition series for K. That is, I can't show that (K & Hi)/(K & Hi-1) is simple.

Surely you mean abelian, not simple?

 

[samkolb]

Thank you for responding to my question.

As I said in the preface to my question, I need to find a composition series for K in which each of the factor groups is abelian. So I need each of the factor groups (K & Hi)/(K & Hi-1) to be both abelian and simple.

 

[gel]

What I can't show is that the distinct groups among (K & Hi) from a composition series for K. That is, I can't show that (K & Hi)/(K & Hi-1) is simple.

Why do you need to show that? K\cap H_{i}/K\cap H_{i-1} is isomorphic to a subgroup of H_i/H_{i-1}, so it is abelian, and this is enough to show that K is simple.

Actually, if K is a strict subgroup of G, you will have H_i\cap K=H_{i-1}\cap K for at least one i. So, K\cap H_i will not be a composition series -- you would have to remove some of them from the series.

 

[gel]

I've got 2 further comments on your question.

First, you don't need to use a composition series. Any subnormal series for which the quotients H_i/H_{i-1} are Abelian will do. Then, you can eliminate any H_i for which H_i=H_{i-1}. Then, if any of the quotients are not simple you can enlarge the series, until you reach a maximum one. This will be a composition series and the quotients will still be abelian.

Second, the only simple abelian groups are the cyclic groups of prime order. Any subgroup will be either the trivial group, or the whole group, so still simple. So, the way you have set up the problem it will be the case that you get a composition series once you eliminate any terms with K\cap H_i=K\cap H_{i-1}.

 

[samkolb]

gel:

Thank you for your comments. They were very helpful.