From the blog
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Let K be an algebraically closed field and a,b n×n matrices over K. Then ab and ba have the same characteristics polynomial.
There are a lot of proof of this fact however here we want to gave a proof that use the machinery of Algebraic Geometry. Recall…
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For which n≥2 there is m∈ℕ such that SO(n)≅RPᵐ ?
In the following if M and N are differentiable manifold we write M≅N iff M and N are diffeomorphic. We know that SO(2)≅S¹≅RP¹ and SO(3)≅RP³…
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Let G be a finite group and φ: G—>G be an automorphism that fixes more than half of the elements of G. Prove that φ is the identity of G.
Let S={g ∈ G : φ(g)=g}. We know that by hypothesis |S|>|G|/2 and, by what was said previously, S is a subgroup of G (it…
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Let G be a finite non-abelian group. Show that |Z(G)| ≤ |G|/4
If by contradiction |Z(G)|>(1/4)|G| then |G/Z(G)|<4 so G/Z(G) is cyclic, so G is abelian, absurd. ■
Marco Damele 