Marco Damele

Tag: teoria dei gruppi

  • If ℤⁿ≅ℤᵐ then n=m.

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    Marco Damele
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    Let ψ:ℤⁿ—>ℤᵐ a group isomorphism. Let’s prove that n=m.Let: H=2ℤⁿ={(2z1,…,2zn) : zi∈ℤ ∀ i=1,..,n} ≤ ℤⁿK= 2ℤᵐ={(2z1,…,2zm) : zi∈ℤ ∀ i=1,..,m} ≤ ℤᵐ Observe that ψ(H)=K. Indeed if (2z1,…,2zn) ∈ H then ψ (2z1,…,2zn) = 2ψ(z1,…,zn). Viceversa if (2z1,…,2zm) ∈ K then ψ⁻¹(2z1,…,2zm)∈H —> (2z1,…,2zm) ∈ψ(H). Now, since ℤᵐ e ℤⁿ are abelian H and…

  • σ-complete groups

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    Marco Damele
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    Consider a finite group G. In general it is interesting to know when, given that N is a maximal normal subgroup of G, we have G ≅ N x G/N. Clearly this does not always happen. For example, take a prime p and G=Zp². The maximal proper normal subgroup is Zp and G/N=Zp but clearly…