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.

Subgroup

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., {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)

From (1) and (2),

a ⋆ e = a ⋆ e’

⇒ e = e’

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

Hence proved.

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)

From (1) and (2),

a ⋆ b = a ⋆ c

⇒ b = c

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

Hence proved.

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.

Let H be any subgroup of G, such that H^-1 = H and HH = H.
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.
H is a subgroup of G if and only if HH^-1 = H.
If H and K are two subgroups of a group G, then HK is a subgroup of G if and only if HK = KH.
If H and K are two subgroups of a group G, then H ⋂ K is a subgroup of G.
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.