# Questions tagged [involutions]

For problems related to involutions , that is functions that are their own inverses .

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### Is there an involutive automorphism mapping two given elements of a poset or lattice?

$S$ is a finite poset or lattice ; $A$ and $B$ two distinct elements. If there is at least one automorphism that maps $A$ to $B$, can I find one such automorphism that is an involution? The set of ...
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### A theorem about the anti-derivation of the algebra

Sadri Hassani wrote down a theorem about the anti-derivation in his Mathematical Physics: A Modern Introduction to Its Foundations - Second Edition. Theorem 3.4.10 Let $\Omega_1$ and $\Omega_2$ be ...
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### Does a fixpoint-free, self inverse homeomorphism define a smooth manifold?

I was wondering if the following theorem still holds if one only requires $\tau$ to be a homeomorphism and not a diffeomorphism: "Let M be a smooth manifold and $\tau:M\to M$ a differentiable function ...
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### what are the eigenvalues of a matrix A of m-involution

"The set of eigenvalues of a matrix A of m-involution (for which $A^m=I$ for an integer m>1) belongs to the set of m-th roots of unity." How do I prove this? It can be shown for m=2, the eigenvalues ...
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### When $\mathbb{C}[u,v]$ is a UFD, for special $u,v \in \mathbb{C}[t]$?

Let $f=f(t) \in \mathbb{C}[t]$ be a non-scalar polynomial. Write $f=a_{2n}t^{2n}+a_{2n-1}t^{2n-1}+\cdots+a_2t^2+a_1t+a_0$, where $a_j \in \mathbb{C}$. The map $\iota: t \mapsto -t$ is an involution ...
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### $\mathbb{C}(s_1,s_2,k)=\mathbb{C}(x,y)$, where $s_1,s_2$ are symmetric and $k$ is skew-symmetric

Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution (= $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ of degree two) defined by $(x,y) \mapsto (x,-y)$. It is not difficult to ...
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### Locus of fixed points of an involution in a surface

I am reading Guletskii Paper "Bloch Conjecture for surfaces with involution and of p_g=0" and I do not know why the following is true. If S is a minimal smooth projective surface with an involution i....
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### Concerning $\mathbb{C}(s_1,s_2,s_3,y)=\mathbb{C}(x,y)$, where $s_1,s_2,s_3$ are symmetric

Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution on $\mathbb{C}[x,y]$ defined by $(x,y) \mapsto (x,-y)$. Let $s_1,s_2,s_3 \in \mathbb{C}[x,y]$ be three symmetric elements with ...
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### The form of $w \in \mathbb{C}(x,y)$ satisfying $\beta(w)=w$, $\beta$ an involution on $\mathbb{C}[x,y]$

The symmetric case: Let $w \in \mathbb{C}(x,y)$, and write $w=\frac{u}{v}$, where $u,v \in \mathbb{C}[x,y]$. Let $\beta: (x,y) \mapsto (x,-y)$; $\beta$ is an involution (= an automorphism of order ...
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### Banach subalgebras and * subalgebras of a C* algebra

Given a unital C* algebra $A$, or more specifically, $A:=\mathcal L(H)$ be the bounded linear operators of a certain Hilbert space $H$. How do its subalgebras look like if one only considers the pure ...
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### Why Exactly 3 Involutions Are Used in The Proof of Roger Heath-Brown?

In the book "Proofs from THE BOOK" (click here, on page 20)by Martin Aigner and Günter M. Ziegler, a proof of representing number as sum of two square is given. The proof is due to Roger Heath-Brown (...
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### Proof of Roger Heath-Brown: Interchanging Solutions of Involution

On page 20 of "Proofs from THE BOOK" (click here) a proof of representing number as sum of two square is given. The proof is due to Roger Heath-Brown (1971,appeared in 1984). The first part of the ...
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### Relationship between trace and determinant of a non-diagonal $3 \times 3$ involutory matrix [closed]

Let $A$ be a $3 \times 3$ real, non-diagonal matrix with $A^{-1}=A$. Show that $\mbox{tr}(A) = -\det(A) = \pm 1$.
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### Conjugacy class of involutions in finite simple group

I've been working on this problem, but I can't seem to do it. For a finite simple group $G$, is there a unique conjugacy class consisting of involutions? Clearly, a conjugacy class with an ...
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### Is there a term for a function that is like an involutory function, but generalized to more recursions?

So, an involutory function is a function that is its own inverse. In other words, a recursive call to this function would return the original input: $$f(f(x)) \equiv x$$ Suppose we had a function ...
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### Does anyone know what $H'_{m}$ means in this group theory paper by Miller?

https://www.jstage.jst.go.jp/article/tmj1911/17/0/17_0_88/_pdf Page 90, third paragraph, where it says "By continuing this process we arrive at a group $H_m$ containing the central of G and each ...
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### Archimedean spiral $\oplus$ sawtooth = circles

Archimedean spiral $\oplus$ sawtooth = circles But what about the involute of the circle instead of the Archimedean spiral? Let the unit circle be given by \gamma_\text{circ}(t) = (x_\text{circ}(t),...
I am reading the textbook "Introduction to Modern Algebra, Joyce 2017" and in the Cyclic groups and subgroups section, there is a following sentence about involution. An involution $a$ is an ...
With exception of the zero matrix, can a matrix be nilpotent $(A^k=0)$ and idempotent $(A^2=A)$ at the same time? and With exception of the identity matrix, can a matrix be idempotent and involutory ...