Waste less time on Facebook — follow Brilliant.
×

Roots of Identity Matrix

I was reading through some problems based on permutations. Particularly, the following problem interested me.

Let \(\mathbb{Z}_n\) denote the set of integers \(\{1,2,\ldots,n\}\) for any positive integer \(n\) and let \(f : \mathbb{Z}_n \rightarrow \mathbb{Z}_n\) be a bijective function defined on \(\mathbb{Z}_n\).

Given that \(n\) is finite, does integer \(k^*\) exist, such that it is always possible to find a \(k \le k^*\) for which \(f^k(x)=x, \forall x \in \mathbb{Z}_n\)?

I suspect that this would turn out to finding the primitive real roots of the identity matrix. That is finding the maximum value of \(k\) such that \(X^k=I_n\) has a \(n \times n\) matrix solution \(X\) , but, \(X^i \neq I_n\) for any \(i < k\)

Taking into account that any arbitrary permutations could consist of \(m\) cycles of length \(i_1,i_2,\cdots,i_m\) with \(i_1+i_2+\cdots+i_m=n\). The problem now boils down to finding the maximum value of the LCM of \(i_1,i_2,\cdots,i_m\) over all possible values of \(m\) i..e,

\[ k^* = \max_{\substack{m=1\cdots n\\ i_1+i_2+\cdots+i_m=n}} LCM (i_1,i_2,\cdots,i_m) \] But, am at a loss in actually proceeding much further.

Note by Janardhanan Sivaramakrishnan
1 year, 5 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...