Examples
The group of units U(Z9) is the group of all integers relatively prime to 9 under multiplication mod 9 (U9 = {1, 2, 4, 5, 7, 8}). All arithmetic here is done modulo 9. Seven is not a generator of U(Z9), since
while 2 is, since:
On the other hand, for n > 2 the symmetric group of degree n is not cyclic, so it is not generated by any one element. However, it is generated by the two permutations (1 2) and (1 2 3 ... n). For example, for S3 we have:
- e = (1 2)(1 2)
- (1 2) = (1 2)
- (1 3) = (1 2)(1 2 3)
- (2 3) = (1 2 3)(1 2)
- (1 2 3) = (1 2 3)
- (1 3 2) = (1 2)(1 2 3)(1 2)
Infinite groups can also have finite generating sets. The additive group of integers has 1 as a generating set. The element 2 is not a generating set, as the odd numbers will be missing. The two-element subset {3, 5} is a generating set, since (−5) + 3 + 3 = 1 (in fact, any pair of coprime numbers is, as a consequence of Bézout's identity).
Read more about this topic: Generating Set Of A Group
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