Examples of Subgroups

Examples 1: The set ℤ of all integers is a subgroup of the set of rational numbers under the operation of addition.

Sol: Since ℤ ⊆ ℚ and ℤ ≠ Φ where <ℚ, +> is a group.

         Also <ℤ, +> forms a group.

         ⇒ <ℤ, +> is a subgroup of group <ℚ, +>.

         
Example 2: A subset H of a group G is a subgroup if and only if

   (i) H is closed under the operation on G.                        That is, if a, b ∈ H,then a, b ∈ H,
   (ii) e ∈ H,

  (iii) H is closed under the taking of inverses.                   That is, if a, b ∈ H, then a^-1∈ H.

Proof: The left implication follows directly from the group axioms and the definition of subgroup. 

For the right implication, we have to verify each group axiom for H. 

Firstly, since H is closed, it is a binary structure, as required, and as mentioned, H inherits associativity from G. In addition, H has the identity element and inverses, 

so H is a group, and we are done.

There is, however, a more effective method. Each of the three criteria listed above can be condensed into a single one.

Example 3:  Let G be a group. Then a subset H ⊆ G is a subgroup is and only if a, b ∈ H ⇒ ab^-1∈ H.

Proof: Again, the left implication is immediate. 

For the right implication, we have to verify the (i)-(iii) in the previous theorem. 

First, assume a ∈ H. 

Then, letting b = a, we obtain aa^-1 = e ∈ H, taking care of (ii). 

Now, since e, a ∈ H we have

                                     ea^-1 = a^-1 ∈ H 

so H is closed under taking of inverses, satisfying (iii). 

Lastly, assume a, b ∈ H. 

Then, since b^-1∈ H, we obtain

                                    a(b^-1)^-1 = ab ∈ H, 

so H is closed under the operation of G, satisfying (i), and we are done.

All right, so now we know how to recognize a subgroup when we are presented with one. Let's take a look at how to find subgroups of a given group. The next theorem essentially solves this problem.

Example 4:  Let G be a group and g ∈ G. Then the subset {gⁿ | n ∈ ℤ  }  is a subgroup of G, denoted <g> and called the subgroup generated by g. In addition, this is the smallestsubgroup containing g in the sense that if H is a subgroup and g ∈ H, then <g> ≤ H.

Proof: First we prove that <g> is a subgroup. 

To see this, note that if h, k ∈ H, then there exists integers n, m ∈ ℤ 

such that h = gⁿ , k = g ^m .. 

Then, we observe that hk^-1 = gⁿg^-m = g^n-m ∈ <g>

 since n – m ∈ ℤ,

so <g> is a subgroup of G, as claimed.

To show that it is the smallest subgroup containing g,

observe that if H is a subgroup containing g, then by closure under products and inverses, gⁿ∈ H  for all n ∈ G.

In other words, <g>⊆ H .

Then automatically <g> ≤ H since <g> is a subgroup of G.