Diploma/Certificado de qualificação profissional de nivel 3 Mathematics (Algebra (MATH311)) Slide Set on §3 Cyclic Groups (Proofs)[not finished], created by Amy Wogan on 20/03/2016.
i) aH = bH ⟺ ah = b for some h∈Hii) ~ is an equivalence relationiii) aH = [a] = {a'∈G : a'~a}Proof (i):⇒: Since b = b∙1, 1∈H, b∈bH,bH = aH (by assumption) ⟹ b∈aH ⟹ b = ah for some h∈H⇐: Suppose ah = b. Show bH ⊂ aH.For any bh'∈bH (h'∈H), bh' = (ah)h' = a(hh') ∈ aHbecause hh'∈H ⟹ bH⊂aHShow aH⊂bH.ah = b ⟹ a = bh−1By a similar argument, aH⊂bH.Together, aH = bH.∎
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Proposition 3.1:
Proof (ii):
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Proposition 3.1:
Proof (iii)
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Theorem 3.1: Lagrange theorem
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Proposition 3.3:
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Theorem 3.2
A cyclic group is abelian.Proof:
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Theorem 3.3:
A subgroup of a cyclic group is cyclic.Proof:
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Corollary 3.1:
The subgroups of Z are exactly the groups nZ for n∈Z.Proof:The cyclic subgroup of Z are exactly those of the form ⟨n⟩
for n∈Z. they are easily seen to be exactly he groups nZ for n∈N*. By the above theorem (3.3) they are the only subgroups of Z∎
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Theorem 3.4:
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Corollary 3.2:
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Theorem 3.5:
The group Z is the only infinite group, up to isomorphism.
