orbit {permutations} | R Documentation |
Orbits of integers
Description
Finds the orbit of a given integer
Usage
orbit_single(c1,n1)
orbit(cyc,n)
Arguments
c1 , n1 |
In (low-level) function |
cyc , n |
In (vectorized) function |
Value
Given a cyclist c1
and integer n1
, function
orbit_single()
returns the single cycle containing integer
n1
. This is a low-level function, not intended for the
end-user.
Function orbit()
is the vectorized equivalent of
orbit_single()
. Vectorization is inherited from
cbind()
.
The orbit of a fixed point is
; the code uses an ugly hack to ensure that
this is the case.
Note
Orbits are useful in a more general group theoretic context. Consider
a finite group acting on a set
, that is
Writing , we define
to be a group action if
and
where
and
is the group identity. For any
we
define the orbit
of
to be the set of
elements of
to which
can be moved by an element
of
. Symbolically:
Now, we abuse notation slightly. In the context of permutation
groups, we consider a fixed permutation . We
consider the group
,
that is, the group generated by
; the
group action is that of
on set
with the obvious definition
for
and
. This clearly is a group action as
and
.
Expressing in cycle form makes it easy to see
that the orbit of any element
of
is just the
other members in the cycle containing
. For example,
consider
and
.
Then
Because we are only interested in the effects on , we only
need to consider the cycle
: this is the only cycle that
affects
, and the
cycle may be ignored as it
does not affect element 4. So
[observe the slight notational abuse above: “”
means “the function
with
,
, and
”].
Author(s)
Robin K. S. Hankin
See Also
Examples
orbit(as.cycle("(123)"),1:5)
orbit(as.cycle(c("(12)","(123)(45)","(2345)")),1)
orbit(as.cycle(c("(12)","(123)(45)","(2345)")),1:3)
data(megaminx)
orbit(megaminx,13)