This included, as a by-product, the classification of all minimal finite simple groups ( simple groups for which every proper subgroup is solvable ).
12.
This proves that every finite group of odd order is solvable, as a minimal counterexample must be a simple group such that every proper subgroup is solvable.
13.
For example, any non-identity proper subgroup of the nonabelian group of order six is a maximal nilpotent subgroup, but only those of order two are Carter subgroups.
14.
There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup : " O h " and " D 6h ".
15.
This same paper gives a number of examples of groups which cannot be realized as Zappa Sz�p products of proper subgroups, such as the quaternion group and the alternating group of degree 6.
16.
The Feit Thompson theorem can be thought of as the next step in this process : they show that there is no non-cyclic simple group of odd order such that every proper subgroup is solvable.
17.
Note that in the classification of finite simple groups, " K "-group is more used to mean a group whose proper subgroups only have composition factors amongst the known finite simple groups.
18.
A famous class of counterexamples to Burnside's problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite cyclic group, the so-called Tarski Monsters.
19.
But a finite group ( the multiplicative group of " D " in our case ) cannot be a union of conjugates of a proper subgroup; hence, " n " = 1.
20.
This notion may also be considered as a numerical range " relative " to the proper subgroup U ( K ) \ times U ( M ) of the full unitary group U ( KM ).
