From the blog
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Is there an infinite group with a finite number of subgroups?
Suppose that G is an infinite group with a finite number of subgroups. We have two possibilities, G contains an infinite cyclic subgroup or it…
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Let G be a finite group such that Aut(G)={e}. Prove that G is isomorphic to ℤ₂
Since Aut(G)={e} we would have that G is abelian (because we would have G/Z(G) ={e} and so G=Z(G) ) then the inversion x|—>x⁻¹ is an…
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Let G be a finite group. Is there always a finite group H such that G≅Aut(H)?
This is false in general. Indeed we will prove that if m is an odd positive integer then doesn’t exist a finite group H such…
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Solve in ℤ₅ the equation x⁷ + x³ – x +1= 0.
If x ∈ ℤ₅ such that x⁷+x³-x+1=0 then x(x⁶+x²-1)=-1 i.e. x(2x²-1)=4. Now (when I write x=m i mean the class of m modulo 5) x=0…
Marco Damele 