Understanding Subgroups in Group Theory

In group theory, a subgroup is a subset of a group that is itself a group under the same operation. While it is relatively straightforward to identify subgroups that satisfy the group criteria, sometimes subsets do not fulfill these requirements. This article explores examples of subsets that are subgroups but specifically fail to be groups, illustrating the criteria needed for a subset to be considered a group.

Non-empty Subset Without an Identity Element

Consider the group ( G mathbb{Z} ) (the integers under addition) and the subset ( H {2, 4, 6, 8, ldots} ) (the positive even integers). This subset no longer satisfies the group property because it lacks the identity element ( 0 ). In a group, the identity element is required for every element to find a corresponding inverse, and as ( 0 otin H ), ( H ) cannot form a group. This is the most basic way a subset can fail to be a group.

Non-closed Subset

Next, consider the group ( G mathbb{Z}_6 ) (the integers modulo 6 under addition) and the subset ( H {1, 2, 3} ). This subset does not form a group because it is not closed under the operation of addition modulo 6. For instance, ( 1 2 3 ), which is in ( H ), but ( 2 2 4 otin H ). This violation of closure is a significant reason why ( H ) cannot be a group.

Subset Without Inverses

In the group ( G S_3 ) (the symmetric group on 3 elements), consider the subset ( H {1, 1, 2} ). While the identity element ( 1 ) is present, ( 2 ) lacks an inverse in ( H ). For a subset to be a group, every element must have an inverse within the subset. Since the operation on ( 2 ) does not yield another element in ( H ) when its inverse is taken, ( H ) fails to meet this criterion.

Finite Subset of an Infinite Group

Examining the example ( G mathbb{R} ) (the real numbers under addition) and the subset ( H {1, 2, 3} ), we see that this subset is not closed under addition. Although ( 1 2 3 ) is in ( H ), ( 1 3 4 otin H ). This violation of closure is another reason why ( H ) fails to be a group. Additionally, the identity element ( 0 otin H ), which is also necessary for a group.

Subset Lacking Closure or Identity

The subset ( H {1, 3} ) in the group ( G mathbb{Z}/4mathbb{Z} ) (integers modulo 4) fails to be a group because it does not contain the identity element ( 0 ) and is not closed under the group operation. For example, ( 1 1 2 otin H ), demonstrating a lack of closure. This additive property is crucial for any subset to be considered a group.

Summary of Group Properties

To be considered a group, a subset must satisfy the following properties:

It must be non-empty. It must be closed under the group operation. It must contain the identity element of the group. Every element in the subset must have an inverse within the subset.

Violating any of these properties means the subset fails to form a group. Understanding these examples and the properties of groups helps in recognizing subgroups that do not qualify as groups.