Cyclic Group

Definition (Cyclic Groups) Let be a group with If the set of integral powers of is a subgroup of with respect to then it is called the subgroup generate by and is denoted by

Definition (Order of an Element) Let then the smallest positive integer such that is called the order of the element If there is no such integer, then we say that has infinite order. The order of an element is denoted or sometimes

Proposition (Cyclic Subgroups) If is a group and then  the set of all integral powers of is a subgroup of and is denoted by Further, if , then and

    Proof. The set of integral powers of is non-empty since   If then and for some integers and Then as desired.
    If then for some integer Thus, If and using the Division Algorithm, where it follows that Since is the least positive integer such that as desired.
    First notice that has distinct elements. To see this suppose with then and so Since is the least positive integer with it follows that   Thus, and so the elements in are distinct. Finally, consider an integral power of say then by the Division Algorithm, with and so as desired.

Definition (Cyclic Groups) Let be a group with Then denotes the intersection of all subgroups of that contains We say that generates or that is generated by ; and write when

Proposition (Group Generated by a Subset) ) Let be a group with Then is the unique smallest subgroup of that contains in the sense that is a subgroup containing and if is any subgroup of that contains then contains Further, if is also a subset of , then if and only if   and

    Proof. Since any intersection of subgroups is a subgroup, is a subgroup of Also, if then by definition of intersection of sets. So is a subgroup of which contains If is a subgroup of containing and then since is the intersection of all subgroups of containing Therefore, if is any subgroup of containing then contains The set is unique in the following sense, if is any subgroup of also satisfying the hypothesis, then it follows that and also Finally, let be a subset of and assume then and since and Conversely, if    then and if   then Therefore, as desired.

Group Theory Books

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