§ 3.3 Constructing Examples: Lab Questions

Solution: Let H be a subgroup of an abelian group G. If x belongs to H and g belongs to G, then gxg^-1 = gg^-1x = x belongs to H.

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2. Let G₁ and G₂ be groups, and let G be the direct product G₁ × G₂.
Let H = { (x₁,x₂) in G₁ × G₂ | x₂ = e } and
K = { (x₁,x₂) in G₁ × G₂ | x₁ = e }.
Prove that H and K are normal subgroups of G.
Note: Exercise 3.3.9 in the text shows that H and K are subgroups for which HK = G and HK = {e}.

Solution: If (x₁,e) is any element of H, and (g₁,g₂) is any element of G₁ × G₂, then

(g₁,g₂) (x₁,e) (g₁,g₂) ^-1 = (g₁,g₂) (x₁,e) (g₁ ^-1,g₂ ^-1)
      = (g₁x₁g₁ ^-1, g₂ e g₂ ^-1) = (g₁x₁g₁ ^-1,e).

Since (g₁x₁g₁ ^-1,e) belongs to H, this shows that H is a normal subgroup of G. A similar argument shows that K is also a normal subgroup of G.

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