After reading https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n, I have a question regarding how to find the generator. For instance, take the group {1, 2, 4, 5, 7, 8} for $ Z /9\mathbb Z$, would the order be 6 (since there are 6 elements) and the exponent be 9, making this a non-cyclic group? Thanks!
Order and exponent for group of invertible residues modulo n under multiplication
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abstract-algebra
group-theory
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0This group is cyclically generated by 2, so it has exponent 6. On the other hand, if you look at $\mathbb{Z}_8^\times$, the group has order 4, but every non-identity element has order 2, so the exponent is 2. – 2017-02-14
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0So the order being the lcm of all elements correct? I'm just a bit confused: does order for $Z/nZ$ work the same as regular integer groups? – 2017-02-14
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0The order of the group is the number of elements it contains, e.g. $\mathbb{Z}_8^\times=\{1,3,5,7\}$ has order 4. The order of an element is the size of the cyclic subgroup it generates, e.g $|3|=2$ since $\langle 3\rangle=\{3,1\}$ (note $3^2=9=1$ in $\mathbb{Z}_8$). The exponent of a group is the maximal order of an element of the group. – 2017-02-14
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0The cardinality of ${\Bbb Z}_n^*$ is the famous **EulerPhi** arithmetic function. – 2017-02-14