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Unformatted text preview: 96 2. BASIC THEORY OF GROUPS Definition 2.2.10. Let a be an element of a group G . The set h a i D f a k W k 2 Z g of powers of a is called the cyclic subgroup generated by a . If there is an element a 2 G such that h a i D G , we say that G is a cyclic group. We say that a is a generator of the cyclic group. Example 2.2.11. Take G D Z , with addition as the group operation, and take any element d 2 Z . Because the group operation is addition, the set of powers of d with respect to this operation is the set of integer multiples of d , in the ordinary sense. For example, the third power of d is d C d C d D 3d . Thus, h d i D d Z D f nd W n 2 Z g is a cyclic subgroup of Z . Note that h d i D h d i . Z itself is cyclic, h 1 i D h 1 i D Z . Example 2.2.12. In Z n , the cyclic subgroup generated by an element OEd is h OEd i D f OEkd W k 2 Z g D f OEkOEd W OEk 2 Z n g . Z n itself is cyclic, since h OE1 i D Z n ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08