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Are all groups of order $156$ solvable?

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Yep. The only non-abelian simple group of order less than or equal to 156 has order 60, which doesn't divide 156. But this is worth proving directly from the Sylow theorems. Is this homework? –  Qiaochu Yuan Dec 14 '10 at 6:32
    
Yep, it is.. I wasnt looking for an answer, more of a definition of solvable and how to tackle it. –  picakhu Dec 14 '10 at 6:35
    
The definition is on Wikipedia (en.wikipedia.org/wiki/Solvable_group). Are you familiar with examples where the Sylow theorems are used to classify the groups of order n for some reasonably small n? –  Qiaochu Yuan Dec 14 '10 at 6:38
    
I came across ca.answers.yahoo.com/question/index?qid=20091128130200AAPZVBY and understand about 50-60% of it. –  picakhu Dec 14 '10 at 6:40
    
@Qiaochu Yuan : What would be a preferred refrences for sylow theorems? –  Arjang Dec 14 '10 at 6:47

1 Answer 1

up vote 3 down vote accepted

Ask GAP:

gap> AllSmallGroups(156, IsSolvable, false);
[  ]
gap> 

This gives the list of all groups of order 156 which are not solvable.

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