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November 30th, 2011, 12:40 PM  #1 
Senior Member Joined: Nov 2011 Posts: 100 Thanks: 0  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 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! 
December 19th, 2011, 05:22 PM  #2  
Senior Member Joined: Nov 2011 Posts: 100 Thanks: 0  Re: Characterization of nonsolvable groups Quote:
 
December 30th, 2011, 07:25 AM  #3 
Senior Member Joined: Sep 2008 Posts: 150 Thanks: 5  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. 
January 3rd, 2012, 05:23 PM  #4 
Senior Member Joined: Nov 2011 Posts: 100 Thanks: 0  Re: Characterization of nonsolvable groups
thank you for your response!


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