8.1 Products and Factors
1. a. XY = {τ, τσ}{τ, τσ2} = {τ2, τ2σ2, τστ, τστσ2} = {ε, σ2, τ2σ2, τ2σ4} = {ε, σ2, σ} . YX = {τ, τσ2}{τ, τσ} = {τ2, τ2σ, τσ2τ, τσ2τσ} = {ε, σ, τ2σ, τ2σ2} = {ε, σ, σ2}.
3. If G′ ⊆ H, then H/G′ 
G/G′ because G/G′ is abelian. Hence H G by the correspondence theorem.
G/G′ because G/G′ is abelian. Hence H G by the correspondence theorem.
4.
a. G = D6 = {1, a, . . ., a5, b, ba, . . ., ba5}, o(a) = 6, o(b) = 2, aba = b. We have K = Z(D6) = {1, a3}. Write 
, 
. Then 
, 109 
, 
, 
. Hence G/K ≅ D3 and the only subgroups of G/K are 
and G/K. So the subgroups of G containing K are G, K and
, . Then , 109 , , . Hence G/K ≅ D3 and the only subgroups of G/K are and G/K. So the subgroups of G containing K are G, K and
Note that H1 ≅ H2 ≅ H3 ≅ K4 —the Klein group (in contrast with ...
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