Subgroups

In mathematics, group theory is one of the most important branches, where we learn about different algebra concepts, such as groups, subgroups, cyclic groups, and so on. As we know, a group is a combination of a set and a binary operation that satisfies a set of axioms, such as closure, associative, identity and inverse of elements. A subgroup is defined as a subset of a group that follows all necessary conditions to be a group. Let’s understand the mathematical definition of a subgroup here.

Subgroups Definition

Let (G, ⋆) be a group and H be a non-empty subset of G, such that (H, ⋆) is a group then, “H” is called a subgroup of G.

That means H also forms a group under a binary operation, i.e., (H, ⋆) is a group.

Also, any subset of a group G is called a complex of G.

Below are some important points about subgroups.

• A subset H of a group G is a subgroup of G, if H itself is a group under the operation in G.
• A subgroup of a group consisting of only the identity element, i.e., is called the trivial subgroup.
• A subgroup H of a group G, a proper subset of G, i.e., H ≠ G is called the proper subgroup and is represented by H < G. This can be read as “H is a proper subgroup of G”.
• If H is a subgroup of G, then G may be called an over group of H in some cases.

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Theorems on Subgroups

Theorem 1:

H is a subgroup of G. Prove that the identity element of H is equal to the identity element in G.

Proof:

Given that H is a subgroup of G.

Let us assume that e and e’ be the two identity elements in H and G, respectively.

Let a ∈ H ⇒ a ∈ G [since H is a subset of G]

Identity element in group H = e

Thus, a ⋆ e = e ⋆ a = a…..(1)

Identity element in group G = e

Therefore, a ⋆ e’ = e’ ⋆ a = a…..(2)

That means, the identity element in H is equal to the identity element in G.

Theorem 2:

H is a subgroup of G. The inverse of any element in H is equal to the inverse of the same element in G.

Proof:

Given that H is a subgroup of G.

Consider a ∈ H ⇒ a ∈ G

Let us assume that b and c are two inverse elements of a in H and G respectively.

Let b be the inverse element of a in H.

Then, a ⋆ b = b ⋆ a = e….(1)

Let c be the inverse element of a in G.

Then, a ⋆ c = c ⋆ a = e….(2)

That means the inverse element of a in H is equal to the inverse element of a in G.

Difference between Groups and Subgroups

The below table illustrates a few differences between groups and subgroups.

Group

Subgroup

A group is a set combined with a binary operation, such that it connects any two elements of a set to produce a third element, provided certain axioms are followed.

A subgroup is a subset of a group.

H is a subgroup of a group G if it is a subset of G, and follows all axioms that are required to form a group.

Groups satisfy the following laws:

• Closure
• Associative
• Identity element
• Inverse law

Subgroups also satisfy the following laws:

• Closure
• Associative
• Identity element
• Inverse law

The number of elements of a finite group is called the order of a group.

A subgroup is also a group, and the order of a subgroup is less than the order of a group.

Properties of Subgroups

We can also prove the following statements using the properties of groups and subgroups.

1. Let H be any subgroup of G, such that H -1 = H and HH = H.
2. H is a non-empty complex of a group G. The necessary and sufficient condition for H to be a subgroup of G is: a, b ∈ H ⇒ ab -1 ∈ H, where b -1 is the inverse of b in G.
3. H is a subgroup of G if and only if HH -1 = H.
4. If H and K are two subgroups of a group G, then HK is a subgroup of G if and only if HK = KH.
5. If H and K are two subgroups of a group G, then H ⋂ K is a subgroup of G.
6. The union of two subgroups of a group is a subgroup, if and only if one is contained in the other. (or) If H and G are two subgroups of G, then H U K is a subgroup of G, if and only if H ⊆ K or K ⊆ H.

Frequently Asked Questions on Subgroups – FAQs

What is the definition of a subgroup?

A subgroup is a subset of a group that itself is a group. That means, if H is a non-empty subset of a group G, then H is called the subgroup of G if H is a group.

What makes a subset a subgroup?

A subset of a group is said to be a subgroup if it holds all group axioms, i.e. associativity, closure, inverse, and identity law under the binary operation of the group.

How many subgroups can a group have?

The number of subgroups of a group can be determined based on the order of a group.