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., {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.

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.

Properties of Subgroups

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