The analogous statement is not in general true for groups of even order. View other such facts for finite groups This fact is useful in work leading up to the Classification of finite simple groups Contents.
One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page. Categories : Group property implications Facts about odd-order groups that break down for even-order groups Facts useful for the Classification Problem. Navigation menu Personal tools Log in. Namespaces Page Discussion.
Suppose 1 holds, i. Then, we want to show that 2 holds. Consider a minimal counterexample to 2 , i.
The complete proof which forms the subject of a page paper is beyond the scope of this page. However, we will attempt to describe the key idea. The idea is to attempt to construct a "minimal counterexample" to 1 , i. Any such minimal counterexample must in particular be a minimal simple group : every proper subgroup is solvable note therefore that the classification of finite minimal simple groups would settle this question; however, such a classification is itself conditional to first proving this fact, hence it does not help.
In particular, it is a N-group : every local subgroup the normalizer of a nontrivial solvable subgroup is solvable.
Or equivalently, for every prime , every p-local subgroup is solvable this follows from the fact that local subgroup of finite group is contained in p-local subgroup for some prime p. An example that is relatively easy to follow is the proof that odd-order and CN implies solvable.
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A CN-group is a group where the centralizer of every non-identity element is nilpotent. The structure of CN-groups allows us to define an equivalence relation on the prime divisors of the group order based on commuting see commuting of non-identity elements defines an equivalence relation between prime divisors of the order of a finite CN-group. Georges Gonthier. Explore further.
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