Cyclic Groups and Subgroups in abstract algebra.ppt
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This document discusses properties of cyclic groups and subgroups of cyclic groups. It provides examples and proofs of theorems regarding cyclic groups. Specifically, it states that a subgroup of a cyclic group is cyclic, the order of any subgroup of a cyclic group must divide the order of the group, and for each positive integer k dividing the group order n, the group has exactly one subgroup of order k. It also provides examples of finding the order of elements in cyclic groups and calculating the Euler phi function for various numbers.
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Cyclic Groups and Subgroups in abstract algebra.ppt
- 1. Chapter 4: Cyclic Groups Properties of Cyclic Groups Classification of Subgroups of Cyclic Groups
- 4. More Examples Therefore, U(8) is not cyclic.
- 9. If |a|=6
- 11. Proof; continue
- 12. Example:
- 19. A subgroup of a cyclic group is cyclic. If |<a>|=n, then the order of any subgroup of <a> divides n. For each +ve integer k, where k divides n, the group <a> has exactly one subgroup of order k, namely, k n a
- 25. Example
- 29. Euler phi function
- 31. Example Find the number of elements of order 8 in the cyclic group . and , Z , 320 8 8000000 Z Z
- 33. Example: find the euler function value for each of the following numbers: n=81, n=100 ) ( find to How n ) ( ) ( ) ( then prime, relatively are n m, If ) ( , number prime any For 1 n m mn p p p p n n n