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§ 3.3 Constructing Examples: Lab Questions

1. Show that every subgroup of an abelian group is a normal subgroup.

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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