Subgroups of Groups

A subset of a group which is itself a group with respect to the same operation is called a subgroup. In this topic, the center and centralizer of a group are shown to be subgroups. Basically, the center of a group is the collection of elements in the group that commute with all elements in the group and the centralizer of a given element in the group is the collection of all elements in the group that commute with that given element. Of course, if the center is the whole group, then the group is Abelian; and the intersection of all the centralizers of the group is the center of the group. Many subgroup tests are detailed including the infamous “fastest gun” subgroup test.

Definition (Subgroup) A subset of a group is a subgroup of if is itself a group with respect to the operation on

Definition (Center and Centralizer) Let be a group. The center of is the set:

and the centralizer of in is the set:

Proposition (Subgroups) Let be a group with a nonempty subset of

    (i) If is a subgroup of is the identity of and is the identity of then

    (ii)  If is a subgroup of and then the inverse of in is the same as the inverse of in

    (iii) If and are subgroups of then is a subgroup of

    (iv) is a subgroup of if and only if is closed under   and every element of has an inverse in

    (v) (Fastest Gun) is a subgroup of if and only if for all

    (vi) is a subgroup of

    (vii)  If then is a subgroup of and is a subgroup of

    Proof.    (i): Let Then since is the identity on Since is the identity on Then by the cancellation law for ,   as desired.
    (ii): Suppose and that and are the inverses of in and repsectively. If is the identity, then By the cancellation law,   as required.
    (iii): Since is associative on is also associative on The identity on is in and and thus is in Let and be the inverse of in Then is in and and therefore is in By definition is a subgroup of as required.
    (iv): Assume that is closed under and that every element of has an inverse in Then since is closed on it is a binary operation on and since is associative on is also associative on If every element of has an inverse in then it remains to show that   (the identity) is actually in Let then there is a in such that But since is closed and so is a subgroup of Conversely, if is a subgroup of then, by definition of subgroup is closed under and every element of has an inverse in
    (v): If is a subgroup of then is closed under multiplication and all elements have inverses, thus for all Assume that for any Then because Also, since for any we have Finally, if then and so is a subgroup as desired.
    (vi): If then for any Therefore, and so is a subgroup of as desired.
    (vii): Let and to show that is a subgroup of let First note that Thus, Therefore, and so is a subgroup of as desired. If then for any given So and thus is a subgroup of

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