A closed subset of a group is a group

Let (G, ⋅) be a group, and let H ⊂ G. If H is closed under ⋅, then H is a subgroup.

Proof:

Let H have m elements, so that

• H = {a₁, a₂, ..., a_m}

Then

• Ha₁ = {a₁², a₂a₁, ..., a_ma₁}

By the cancellation laws, Ha₁ has m elements as well, and since H is closed and a₁ ∈ H, Ha₁ = H.

Since a₁ &isin H = Ha₁, ∃a_j &isin H such that a₁ = a_ja₁. Therefore, a_j = e (∈H). __(1)

Also, ∃a_k &isin H such that a_ka₁ = a_j. That is, a_k = a₁^-1 &isin H.

Similarly, by looking at Ha_i, for i = 2..m, a_i^-1 &isin H. __(2)

So, H:

• is closed (hypothesis)
• is associative (inherited)
• has identity (1), and
• contains inverses (2)

Therefore, H is a group, and so H is a subgroup of G.