Cayley tables for permutation groups
cayley.RdProduces a nice Cayley table for a subgroup of the symmetric group on \(n\) elements
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))
#> () (1,2,3,4) (1,3)(2,4) (1,4,3,2)
#> () () (1,2,3,4) (1,3)(2,4) (1,4,3,2)
#> (1,2,3,4) (1,2,3,4) (1,3)(2,4) (1,4,3,2) ()
#> (1,3)(2,4) (1,3)(2,4) (1,4,3,2) () (1,2,3,4)
#> (1,4,3,2) (1,4,3,2) () (1,2,3,4) (1,3)(2,4)
## Klein group:
K4 <- as.cycle(c("()","(12)(34)","(13)(24)","(14)(23)"))
names(K4) <- c("00","01","10","11")
cayley(K4)
#> 00 01 10 11
#> 00 00 01 10 11
#> 01 01 00 11 10
#> 10 10 11 00 01
#> 11 11 10 01 00
## 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)
#> () (ab) (ac) (bc) (abc) (acb)
#> () () (ab) (ac) (bc) (abc) (acb)
#> (ab) (ab) () (acb) (abc) (bc) (ac)
#> (ac) (ac) (abc) () (acb) (ab) (bc)
#> (bc) (bc) (acb) (abc) () (ac) (ab)
#> (abc) (abc) (ac) (bc) (ab) (acb) ()
#> (acb) (acb) (bc) (ab) (ac) () (abc)
## Now an example from the onion package, the quaternion group:
if (FALSE) { # \dontrun{
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
} # }