We mentioned last time that if we have a group and a subset (*not* necessarily a subgroup) of elements , then there was a unique minimal subgroup of which contained . This subgroup is the intersection of all subgroups which contain and is denoted . In the case when is a singleton (i.e., contains only one element), say , then we just write for the smallest group containing . Groups generated by a single element like this are referred to as *cyclic*.

Recalling the order of a group is simply the cardinality of the underlying set, we define the order of an element of the group to be the order of the subgroup generated by that element; . Also recall that an alternative way of describing a group generated by a set was to talk about all of the ways to multiply (apply the group operation) to elements of the set. In the case of a cyclic group, where the group is generated by a single element, this means that every element in the group is actually a power of the generator. That is, if , then there exists a such that .

If , then is actually the smallest natural number such that . To see this, suppose there was another number, with . Then we would have for every that , where (by Euclid’s algorithm). This gives . If this were the case, then the group would in fact only have order .

Likewise, if and , then . Again, use Euclid’s algorithm to write where . We have again , but this contradicts the minimality of if , so we conclude and .

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