Elementary Properties of Groups

    This topic defines cyclic subgroups, the order of an element, and details the properties of the order of an element. More generally, subgroups generated by subsets are defined and characterized. The subgroup generated by a subset of the group is the intersection of all the subgroups containing the subset. Finally, we give methods to construct more groups using a given collection of groups by using direct products and related methods.

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

    Proof. The set of integral powers of is nonempty since   If then and for some integers and Then as desired.

Definition (Cyclic Groups) If for some then is called a cyclic group and is called the cyclic subgroup generated by

Proposition (Subgroup Generated by an Element) Assume is a group, and there exists two different integers and such that Then

    (i) there is a smallest positive integer such that
    
    (ii) if is an integer, then if and only if and
    
    (iii) the subgroup generated by is
    
    Proof. (i) Without loss of generality, Then and so by the Well-Ordering Axiom there is a least integer such that
    (ii) 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.
    (iii) 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 intgeral power of say then by the Division Algorithm, with and so as desired.

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 (Properties of the Order of an Element) Let then and    

Proposition (Group Generated by a Subset) Let denote the intersection of all subgroups of that contain a subset of 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 anysubgroup 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.

    We say that generates or that is generated by ; and write when

Proposition (Direct Product) If and are groups, then is a group with respect to the operation defined by for all and The group is called the direct product of and and is Abelian if and only if and are Abelian.

Proposition (Properties of Direct Product) Suppose is a subgroup of and is a subgroup of then is a subgroup of is a subgroup of and is a subgroup of

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Cite this as:
Elementary Properties Of Groups
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/elementary-properties-of-groups.html