direct_product

introduction

%%visits: 3 Creating groups of higher order, from smaller groups. ## intuition direct_products are [[group]]s with a coordinate system instead of just one element. For example G = G₁ × G₂ where (e₁, e₂) is the identity

Theorem ℤ_p × ℤ_q ≡ ℤ_pq⇔ p and q are relatively prime.

proof. Let ℤ_p =  < a> and ℤ_q find the generator and show it’s order

Remember the operation of g₁*_G1g_1^′

If G₁ and G₂ are abelian groups then G₁ × G₂ is abelian

Proof: $\left( g_1,g_2 \right) \left( g_1',g_2' \right) = \left( g_1g_1',g_2g_2' \right) = \left( g_1'g_1,g_2'g_2 \right)= \left( g_1',g_2' \right\left( g_1,g_2 \right) )$ hence proved.

|G₁ × G₂| = |G₁||G₂| as Let g_i ∈ G₁ and h_i ∈ G₂. This is implied from the cartesian product and elements in a group being distinct.

G₁ ≡ G₁ × {e₂} ⊂ G₁ × G₂ Think of it as 1-d slice in a 2d plane! ## rigour Definition 7.1 in the lecture notes.

direct_produc := Let G₁ and G₂ be two groups. Then G = G₁ × G₂ = {(g₁, g₂) : g₁ ∈ G₁, g₂ ∈ G₂} with the multiplication law (g₁, g₂)(g_1^′, g_2^′) = (g₁g_1^′, g₂g_2^′). G₁ × G₂ is called the direct_product of G₁ and G₂. The action is sort of [[dot_product]]. ## exam clinic ## examples 1. Let G₁ = {e₁, a₁} with a₁² = e₁ and G₂ = {e₂, a₂} with a₂² = e₂ since the cardinality of the group is 4, it is isomorphic to the kliens_four group or ℤ₄. Recall that ℤ₄ = ⟨b⟩ with b⁴ = e then ℤ₄ = {e, b, b², b³}, |e| = 1, |b| = 4, |b²| = 2, |b³| = 4 (since 3 and 4 are coprime), Returning to G₁ × G₂ all of its elements are of the form: (a₁ⁿ, a₂^m). (a₁ⁿ, a₂^m)² = (a₁²ⁿ, a₂^2m) = (e₁, e₂).

All non-identity elements in G₁ × G₂ have order 2. G₁ × G₂ is not isomorphic to Z₄ as the elements order should be preserved

So it must be isomorphic to G₁ × G₂ ≡ V₄

2. ℤ₂ × ℤ₃ ≡ ℤ₆ with elements (a, b) with a ∈ {0, 1}, b{0, 1, 2}. If we find an element that generates ℤ₂ × ℤ₃, this would be the first question you should ask, since it makes life a lot easier. It is clear that (1, 1) generates, since 2, and 3 are coprime.

3. ℤ₂ × ℤ₄ Forms a discretised torus.

ℝ² :todo:D₃ is not a direct_product, since ## resources tags :math:

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