permutations (version 1.1-5)

cayley: Cayley tables for permutation groups

Description

Produces a nice Cayley table for a subgroup of the symmetric group on \(n\) elements

Usage

cayley(x)

Value

A square matrix giving the group operation

Arguments

x

A vector of permutations in cycle form

Author

Robin K. S. Hankin

Details

Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group acting on G. In this context it means that if we have a vector of permutations that comprise a group, then we can nicely represent its structure using a table.

If the set x is not closed under multiplication and inversion (that is, if x is not a group) then the function may misbehave. No argument checking is performed, and in particular there is no check that the elements of x are unique, or even that they include an identity.

Examples

## cyclic group of order 4:
cayley(as.cycle(1:4)^(0:3))

## Klein group:
K4 <- as.cycle(c("()","(12)(34)","(13)(24)","(14)(23)"))
names(K4) <- c("00","01","10","11")
cayley(K4)


## S3, the symmetric group on 3 elements:
S3 <- as.cycle(c(
    "()",
    "(12)(35)(46)", "(13)(26)(45)",
    "(14)(25)(36)", "(156)(243)", "(165)(234)"
))
names(S3) <- c("()","(ab)","(ac)","(bc)","(abc)","(acb)")
cayley(S3)


## Now an example from the onion package, the quaternion group:
if (FALSE) {
 library(onion)
 a <- c(H1,-H1,Hi,-Hi,Hj,-Hj,Hk,-Hk)
 X <- word(sapply(1:8,function(k){sapply(1:8,function(l){which((a*a[k])[l]==a)})}))
 cayley(X)  # a bit verbose; rename the vector:
 names(X) <- letters[1:8]
 cayley(X)  # more compact
}