| 1. | PGL is in general a proper subgroup of the collineation group.
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| 2. | The isometries form a proper subgroup of, namely PSU ( 2 ).
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| 3. | For, there are additional conditions, i . e . is then a proper subgroup of.
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| 4. | The Galois group of a polynomial of degree n is S _ n or a proper subgroup of that.
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| 5. | A subgroup of ? is called an " isolated subgroup " if it is a segment and is a proper subgroup.
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| 6. | Hence ? is induced from a character of a proper subgroup of " G " containing " N ."
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| 7. | There exist finite groups other than cyclic groups with the property that all proper subgroups are cyclic; the Klein group is an example.
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| 8. | Every proper subgroup of " G " can be assumed a solvable group, meaning that much theory of such subgroups could be applied.
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| 9. | In other words " H " is a maximal element of the partially ordered set of proper subgroups of " G ".
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| 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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