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 watson November 30th, 2011 12:40 PM

Characterization of nonsolvable groups

Hi! So I was just wondering if someone knew what characterizes finite nonsolvable groups? I was thinking about it, and I think if you have a simple, nonabelian group, it will be nonsolvable.. but is the converse true? I'm guessing not, but I don't know much about this. Clearly nonsolvable implies nonabelian. Would it be enough to say that a group $G$ is not solvable if and only if it is nonabelian and has a simple subgroup perhaps?

I'd appreciate any help here, I'm just kind of shooting in the dark with the above statements. Thank you!

 watson December 19th, 2011 05:22 PM

Re: Characterization of nonsolvable groups

Quote:
 Originally Posted by laurence Would it be enough to say that a group $G$ is not solvable if and only if it is nonabelian and has a simple subgroup perhaps?
So I just want to comment on what I posted above, I'm pretty sure one or both of the above implications is not true or not sufficient, but I'm unsure on how to approach it. Thank you for any input! What I'm really looking for is one or more ways to characterize nonsolvable groups.

 Peter December 30th, 2011 07:25 AM

Re: Characterization of nonsolvable groups

Well clearly if a group contains a nonabelian simple subgroup, it is nonsolveable. But I'm pretty sure that the other implication is wrong: From the definition of solveable one gets without problems: A group is nonsolvable if and only if it contains a subgroup, which has a nonabelian simple quotient. The two extreme cases, where the group contains a nonabelian simple subgroup or where it has a nonabelian simple quotient probably not enough.

Examples tend to be complicated, as most small groups are solveable. I think the symmetric group on 5 Elements should be the smallest nonsolveable group without a nonabelian simple quotient. To see that your original conjecture is not true, I would try to find a nonsplit extension of the alternating group A_5 with a solveable group. With more time i might be able to check that one exits using group cohomology, however, that is not constructive and it would be hard, to write it down.

I would guess that there are examples of nonsolveable groups that have neither a nonabelian simple subgroup nor such a quotient, but examples will be hard to construct.

These are just my thoughts, I hope they help a little. For more specific advise just ask again.

 watson January 3rd, 2012 05:23 PM

Re: Characterization of nonsolvable groups