A proper subgroup is a specific type of subgroup that is strictly smaller than the main group it belongs to.
A subgroup, denoted as H, of a group G is called a proper subgroup if H is a proper subset of G. This means that H contains some, but not all, elements of G, and crucially, H is not equal to G itself (H ≠ G). When H is a proper subgroup of G, it is commonly represented by the notation H < G, which can be read as "H is a proper subgroup of G." In such a relationship, the larger group G may sometimes be referred to as an "overgroup" of H.
Understanding the Components
To fully grasp what a proper subgroup entails, let's briefly define its foundational concepts:
- Group: In mathematics, a group is a set equipped with a binary operation that combines any two of its elements to form a third element, in a way that satisfies four fundamental properties: closure, associativity, identity, and invertibility.
- Subgroup: A subgroup is a subset of a group that itself forms a group under the same operation as the parent group. For instance, the set of even integers under addition is a subgroup of the set of all integers under addition.
- Proper Subset: A set A is a proper subset of set B if A is a subset of B, and A is not equal to B (A ≠ B). This implies that there is at least one element in B that is not in A.
Why "Proper"?
The term "proper" distinguishes these subgroups from what might be considered "improper" subgroups. Every group G is technically a subgroup of itself (G ≤ G). However, since G is not a proper subset of itself (as G = G), G is not a proper subgroup of G. The only other trivial subgroup is the identity subgroup {e}, which contains only the identity element of the group. If the group G has more than one element, then {e} is always a proper subgroup.
Key Characteristics of a Proper Subgroup
- It must be a subgroup of the parent group.
- It must be a proper subset of the main group, meaning it contains strictly fewer elements than the main group (unless both are infinite, in which case it's about not being identical).
- It cannot be the entire group itself.
- It can be the trivial subgroup (containing only the identity element), provided the main group is not the trivial group itself.
Examples of Proper Subgroups
Understanding proper subgroups is often best done through examples:
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Integers and Even Integers:
- Consider the group of integers under addition, denoted as (ℤ, +).
- The set of all even integers, denoted as (2ℤ, +), forms a subgroup of (ℤ, +).
- Since 2ℤ is a proper subset of ℤ (e.g., 1 is in ℤ but not in 2ℤ), (2ℤ, +) is a proper subgroup of (ℤ, +). We write this as 2ℤ < ℤ.
-
Cyclic Group of Order 6:
- Let G = {0, 1, 2, 3, 4, 5} be the group under addition modulo 6.
- Consider the subset H = {0, 2, 4}. This forms a subgroup of G under addition modulo 6.
- Since H is a proper subset of G (e.g., 1 is in G but not in H), H is a proper subgroup of G. So, H < G.
- Another proper subgroup would be K = {0, 3}.
Proper Subgroup vs. Subgroup
The distinction between a general subgroup and a proper subgroup is important for precise mathematical discourse:
| Feature | Subgroup (H ≤ G) | Proper Subgroup (H < G) |
|---|---|---|
| Definition | A subset of a group G that is itself a group under G's operation. | A subgroup H of G where H is a proper subset of G. |
| Equality with G | Can be equal to G (H = G) | Cannot be equal to G (H ≠ G) |
| Trivial Subgroup | The identity subgroup {e} is always a subgroup. | The identity subgroup {e} is always a proper subgroup (unless G={e}). |
| Main Group Itself | G is always a subgroup of G. | G is never a proper subgroup of G. |
| Notation | H ≤ G (general) or H < G (if proper) | H < G |
By understanding proper subgroups, mathematicians can analyze the internal structure of groups more precisely, identifying smaller, self-contained algebraic systems within a larger one.
