nielsen | R Documentation |
Vectorized functionality to implement outer automorphisms of the free group
permsymb_single_X(X,f) permsymb_single_f(X,f) permsymb_vec(X,f) permsymb(X,f) autosub_lowlevel(M,e,S) autosub(X,e,S,automorphism_warning=TRUE)
X,S |
Object of class |
f |
Permutation function |
M |
Single free group element, in two-row matrix form |
e |
Single element to substitute |
automorphism_warning |
Boolean, with default |
In 1924, Nielsen showed that the automorphism group of the free group with basis [x_1,...,x_n] is generated by the following four elementary Nielsen transformations:
switch x_1 and x_2
Cyclically permute x_1,x_2,...,x_n to x_2,...,x_n,x_1
Replace x_1 with x_1^-1
Replace x_1 with x_1 x_2.
The functions documented here give vectorized methods to effect such outer automorphisms, using the permutations package.
Operations 1 and 2 above generate the symmetric group S_n and such
automorphisms are effected by function permsymb()
. Operation
3 is carried out by by flip()
and operation 4 by subsymb()
.
Functions permsymb_single_X()
, permsymb_single_f()
,
permsymb_vec()
and subsymb_lowlevel()
are low-level helper
functions that are not really suited for the end user; use
permsymb()
, (flip)
and subsymb()
instead.
Function permsymb()
is intended to work nicely with the
permutations package; see inst/outer.Rmd
for some
illustrations. The function is not perfect.
Robin K. S. Hankin
Wikipedia contributors. (2018, October 29). “Automorphism group of a free group”. In Wikipedia, The Free Encyclopedia. Retrieved 19:58, January 10, 2019, from https://en.wikipedia.org/w/index.php?title=Automorphism_group_of_a_free_group&oldid=866270661
flip
P <- as.free(c("abc","aba","cc","ca")) autosub(P,"c",as.free("xyz")) flip(P,"c") flip(P,"ac")
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