For problems related to involutions , that is functions that are their own inverses .
Questions tagged [involutions]
223 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 eigenvalues…
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
- 2,520
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…
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
- 8,535
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 of…
user237522
- 7,263
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…
John Cataldo
- 2,729
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
- 297
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
- 56,967
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
- 25,412
8
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
- 523
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
- 25,412
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
- 2,740
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…
snivysteel
- 1,134
6
votes
2 answers
Is $f(a)$ normal in $A\ $?
Fact $:$ Let $K$ be a compact subset of $\mathbb C$ and $\Omega$ be an open set containing $K.$ Then there exists a cycle $\Gamma : = \sum\limits_{j=1}^{n} n_{j} \gamma_{j}$ in $\Omega \setminus K$ such that $\text {ind}_{\Gamma} (z) = 1$ for all…
ACB
- 3,068
6
votes
3 answers
What does it mean for a ring to have an involution? Are there any examples?
A ring $R$ is said to be a ring with an involution if there exists a mapping
$*\colon R \to R$ such that for every $a, b \in R$:
$a^{**} = a$,
$(a + b)^* = b^* + a^*$,
$(ab)^* = b^*a^*$.
Can anyone please explain this definition with an…
Amanda
- 179