Mathematics

Questions tagged [involutions]

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

225 questions
18
votes
6 answers

Prove that an involutory matrix has eigenvalues $\pm 1$

I'm trying to prove that an involutory matrix (a matrix where $A=A^{-1}$ ) has only eigenvalues $\pm 1$ . I've been able to prove that $\det(A) = \pm 1$ , but that only shows that the product of the eigenvalues is equal to $\pm 1$ , not the…
spc38
  • 183
10
votes
1 answer

Involutions of the second type in a division algebra

I'm trying to figure out some details about involutions of division algebra, thought maybe someone here might have a better insight. Let $k$ be a $p$-adic or number field, and let $K=k[\sqrt{\delta}]$ be a non-trivial extension of degree $2$. For…
kneidell
  • 2,540
10
votes
1 answer

Classification of analytic involutions

Does there exist a classification of all analytic involutions? I only know $x\mapsto x$ , $x\mapsto -x$ , $x\mapsto\frac{1}{x}$ , and "shifts" thereof.
Alexey
  • 2,786
10
votes
2 answers

Average number of inversions in an involution

I'm working through some exercises in Sedgewick's Analysis of Algorithms, but I'm stuck on 7.45: Find the CGF for the total number of inversions in all involutions of length $N$ . Use this to find the average number of inversions in an involution. I…
Dragonborn Torinn
  • 101
10
votes
1 answer

$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) )$

Suppose $f\colon \mathbb R\to\mathbb R$ is a strictly decreasing function which satisfies the relation $$f(x+y) + f( f(x) + f(y) ) = f( f( x+f(y) ) + f( y+f(x) ) ) , \quad \forall x , y \in\mathbb R $$ Prove that $f( f(x))=x$
Souvik Dey
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9
votes
0 answers

Ideal of $k[x,y]$ invariant under an involution

Let $k$ be a field of characteristic zero. The two-dimensional case: Let $p,q \in k[x,y]$ , $I=\langle p,q \rangle$ a proper ideal of $k[x,y]$ and $\delta$ an involution on $k[x,y]$ , namely, a $k$ -algebra automorphism of order two. Denote the set…
user237522
  • 7,267
9
votes
1 answer

If $f$ has no non trivial fixed points and $f\circ f$ is the identity then $f(x)=x^{-1}$ and $G$ is abelian for $f$ an automorphism of $G$

Let $f$ be an automorphism of the finite group $G$ such that $f\circ f=id$ and $f(x)=x\implies x=e$ Prove that $f(x)=x^{-1}~\forall x\in G$ If we can prove that $f(x)$ and $x$ commute for any given $x\in G$ then we're done with the proof because it…
John Cataldo
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9
votes
1 answer

Composition of 2 involutions

How can we prove that any bijection on any set is a composition of 2 involutions ? Since involutions are bijections mapping elements of a set to elements of the same set, I find it weird that this applies to any bijection. Thanks for your help !
TedMosby
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8
votes
1 answer

$f(f(z)) = z, f' (f'(z)) = z $?

Consider an analytic function $f(z)$ defined over a nonempty subset of the complex plane such that $f(f(z)) = z$ . (Also called analytic involutions) One wonders when we also have $f'(f'(z)) = z$ where $'$ denotes the derivative with respect to $z$…
mick
  • 18,413
8
votes
0 answers

"Flipping fractions" and a strange involution

Apologies for the vague title. Part of my question is what the technically accurate language is for this function $f$ I'm asking about. The Function I found this map on $\mathbb C^3$ I'd like to learn more about. It takes a triplet of numbers…
Chris Wolird
  • 339
8
votes
1 answer

Transfer and fusion in a centralizer

Suppose $G$ is a finite group of order divisible by $8$ , with an element $\tau$ of order 2 whose centralizer $C_G(\tau)$ is elementary abelian of order 4. I suspect $G/[G,G]$ must have even order, but I'm not sure how to prove it. I thought about…
Jack Schmidt
  • 57,327
8
votes
3 answers

Involution that brings sets to disjoint sets

Let $A$ be a collection of subsets of $\{1,2,\dots,n\}$ that is closed under taking subsets (that is, if $U\in A$ and $V\subseteq U$ then $V\in A$). Is there always an involution $f:A\to A$ such that $f(V)\cap V=\emptyset$ for all $V\in A$? I'm…
Akiva Weinberger
  • 25,754
7
votes
1 answer

Is there a fixed-point free involution of the Möbius strip?

Is there a fixed-point free continuous involution of the Möbius strip? (Meaning a function $f:M\to M$ such that $f\circ f={\rm id}$ and $f(x)\ne x$ for all $x$ .) The Lefschetz fixed-point theorem is useless here because the Möbius band is homotopic…
Akiva Weinberger
  • 25,754
7
votes
0 answers

What categorical property do these forgetful functors have in common?

Consider the following examples: The forgetful functor $U_1: \operatorname{Vect}_\mathbb{C} \to \operatorname{Vect}_\mathbb{R}$ The forgetful functor $U_2: \operatorname{Diff}^{\text{or}} \to \operatorname{Diff}$ (oriented manifolds and…
Turion
  • 2,790
6
votes
4 answers

Order reversing pairing of factors via cofactor involution (reflection)

The sum of the two smallest positive divisors of an integer $N$ is $6$, while the sum of the two largest positive divisors of $N$ is $1122$. Find $N$. I came across this question in a Math Olympiad Competition. I am able to find out that the two…
snivysteel
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