cyclic permutation

cyclic permutation

Let

A

=

{

a

0

,

a

1

,

…

,

a

n

-

1

}

be a

finite set

indexed by

i

=

0

,

…

,

n

-

1

. A

cyclic permutation

on

A

is a

permutation

π

on

A

such that, for some

integer

k

,

π

⁢

(

a

i

)

=

a

(

i

+

k

)

(

mod

n

)

,

where

a

(

mod

b

)

:=

a

-

⌊

a

/

b

⌋

⁢

b

, the

remainder

of

a

when divided by

b

, and

⌊

⋅

⌋

is the

floor function

.

For example, if

A

=

{

1

,

2

,

…

,

m

}

such that

a

i

=

i

+

1

. Then a cyclic permutation

π

on

A

has the form

π

⁢

(

1

)

=

r

π

⁢

(

2

)

=

r

+

1

⋮

π

⁢

(

m

-

r

+

1

)

=

m

π

⁢

(

m

-

r

+

2

)

=

1

⋮

π

⁢

(

m

)

=

r

-

1

.

In the usual

permutation notation

, it looks like

π

=

(

1

2

⋯

m

-

r

+

1

m

-

r

+

2

⋯

m

r

r

+

1

⋯

m

1

⋯

r

-

1

)

Remark

. For every finite set of

cardinality

n

, there are

n

cyclic permutations. Each non-trivial cyclic permutation has order

n

. Furthermore, if

n

is a

prime number

, the set of cyclic permutations forms a

cyclic group

.

Cyclic permutations on words

Given a word

w

=

a

1

⁢

a

2

⁢

⋯

⁢

a

n

on a set

Σ

(may or may not be finite), a

cyclic conjugate

of

w

is a word

v

derived from

w

based on a cyclic permutation. In other words,

v

=

π

⁢

(

a

1

)

⁢

π

⁢

(

a

2

)

⁢

⋯

⁢

π

⁢

(

a

n

)

for some cyclic permutation

π

on

{

a

1

,

…

,

a

n

}

. Equivalently,

v

and

w

are cyclic conjugates of one another iff

w

=

s

⁢

t

and

v

=

t

⁢

s

for some words

s

,

t

.

For example, the cyclic conjugates of the word

a

⁢

b

⁢

a

⁢

b

⁢

a

over

{

a

,

b

}

are

b

⁢

a

⁢

b

⁢

a

2

,

a

⁢

b

⁢

a

2

⁢

b

,

b

⁢

a

2

⁢

b

⁢

a

,

a

2

⁢

b

⁢

a

⁢

b

,

and itself

.

Strictly

speaking,

π

is a cyclic permutation on the

multiset

A

=

{

a

1

,

…

,

a

n

}

, which can be thought of as a cyclic permutation on the set

A

′

=

{

(

1

,

a

1

)

,

…

,

(

n

,

a

n

)

}

. Furthermore,

π

can be extended to a

function

on

A

*

: for every word

w

=

a

ϕ

⁢

(

1

)

⁢

⋯

⁢

a

ϕ

⁢

(

m

)

,

π

⁢

(

w

)

:=

π

⁢

(

a

ϕ

⁢

(

1

)

)

⁢

⋯

⁢

π

⁢

(

a

ϕ

⁢

(

m

)

)

, where

ϕ

is a permutation on

A

.

Given any word

w

=

a

1

⁢

a

2

⁢

⋯

⁢

a

n

on

Σ

, two cyclic permutations

π

1

,

π

2

on

{

a

1

,

…

,

a

n

}

are said to be the

same

if

π

1

⁢

(

w

)

=

π

2

⁢

(

w

)

. For example, with the word

a

⁢

b

⁢

a

⁢

b

, then the cyclic permutation

(

1

2

3

4

3

4

1

2

)

is the same as the

identity

permutation. There is a

one-to-one correspondence

between the set of all cyclic conjugates of

w

and the set of all

distinct

cyclic permutations on

{

a

0

,

a

1

,

…

,

a

n

}

.

Remarks

.

•

In a group

G

, if two

elements

u

,

v

are cyclic conjugates of one another, then they are

conjugates

: for if

u

=

s

⁢

t

and

v

=

t

⁢

s

, then

v

=

t

⁢

(

s

⁢

t

)

⁢

t

-

1

=

t

⁢

u

⁢

t

-

1

.

•

Cyclic permutations were used as a ciphering scheme by Julius Caesar. Given an

alphabet

with letters, say

a

,

b

,

c

,

…

,

x

,

y

,

z

, messages in letters are encoded so that each letter is shifted by three places. For example, the name

“Julius Caesar” becomes “Mxolxv Fdhvdu”.

A ciphering scheme based on cyclic permutations is therefore also known as a

Caesar shift cipher

.

Title

cyclic permutation

Canonical

name

CyclicPermutation

Date of creation

2013-03-22 17:33:54

Last modified on

2013-03-22 17:33:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 94B15

Classification

msc 20B99

Classification

msc 03-00

Classification

msc 05A05

Classification

msc 11Z05

Classification

msc 94A60

Synonym

Caesar cipher

Related topic

CyclicCode

Related topic

SubgroupsWithCoprimeOrders

Defines

Caesar shift cipher

Defines

cyclic conjugate

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