Tag: teoria dei gruppi
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If m is odd and n a natural number (2ᵐ-1,2ⁿ+1)=1.
In the following i denote with (a,b) the greatest common divisor between two integers a and b. Let p now be a possible prime divisor of A=2ᵐ-1 and B=2ⁿ+1. Let [2]_p denote the class of 2 modulo p. Let us note that since (2,p)=1 then [2]_p is an element of the group of invertibles of…
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There is no group G such that G’=D2n, for n≥3.
Let G be a group. We denote with G’ the commutator subgroup of G and with D2n the diedral group of order 2n, with n a natural number ≥3. Let us prove that there is no group G such that G’=D2n. Suppose that such a group exists, then since G’=<r,s> we get: <r> char G’…
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If R is a local ring then R[[x]] is local.
Recall that if S is a commutative ring with unit, S is local if and only the set of non unit of S is an ideal, i.e, denoting with S* the unit of S, S-S* is an ideal of S. Now, let m be the maximal ideal of R. We prove that R[[x]] – R[[x]]*…
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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 is a non-empty subset of G and if x,y∈S we have that φ(x⁻¹y)=φ(x⁻¹)φ(y)=φ(x)⁻¹φ(y) )therefore|G|=|S|k for non-zero natural k. If k=1 I am done. In fact:|G|=|S|k>(|G|/2)k. Therefore if k>1 then k≥2…
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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. ■
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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 does not. If we are in the first case then this subgroup, being isomorphic to ℤ, has in turn an infinite number of subgroups, absurd. Then we must be in…
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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 automorphism of G. If G has an element of order greater then 2 then the inversion is not trivial so G is a 2-elementary abelian groups and so G is…
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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 that ℤm≅Aut(H). Indeed if such a group exists then Aut(H) is cyclic and not trivial, by the last post at link https://marcodamele.blog/2024/09/05/let-g-be-a-group-such-that-autg-is-cyclic-and-not-trivial-then-autg-is-finite-and-his-order-is-even/ we would have that Aut(H) is even which…
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Six proofs that √2 is irrational
Dimostriamo che √2 è irrazionale usando 6 dimostrazioni diverse. 1° Proof: Suppose that √2 ∈ ℚ then there exist two coprime integers a and b such that √2=a/b. It follows that 2b²=a². So a² is even, that is, a is even, or a=2k with k being an integer. But then 2b²=4k², or b²=2k², from which…
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Finite groups of class 2 e 3.
Let G be a finite group. We want to classify all finite groups with exactly two and three conjugacy classes. The number of conjugacy classes of G is called the class of G and is denoted by Cl(G). In the following if x ∈ G the conjugacy class of x is denoted by Cl(x). Then…
Marco Damele 