1. PGL is in general a proper subgroup of the collineation group. 2. The isometries form a proper subgroup of, namely PSU ( 2 ). 3. For, there are additional conditions, i . e . is then a proper subgroup of. 4. The Galois group of a polynomial of degree n is S _ n or a proper subgroup of that. 5. A subgroup of ? is called an " isolated subgroup " if it is a segment and is a proper subgroup . 6. Hence ? is induced from a character of a proper subgroup of " G " containing " N ." 7. There exist finite groups other than cyclic groups with the property that all proper subgroups are cyclic; the Klein group is an example. 8. Every proper subgroup of " G " can be assumed a solvable group, meaning that much theory of such subgroups could be applied. 9. In other words " H " is a maximal element of the partially ordered set of proper subgroups of " G ". 10. It is not possible for a finite group to be mapped by inclusion ( i . e . by a monomorphism ) as a proper subgroup of itself.
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