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0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 
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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 49 "                                                 " }{TEXT 256 
26 "   Finite groups in MAPLE5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 34 "symm_gp0.mws, 11-97 and 12-99, wdj" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "This work
sheet contains programs to " }}{PARA 0 "" 0 "" {TEXT -1 83 "          \+
        1. list all the elements of a permgroup as disjoint cycles usi
ng " }{TEXT 285 12 "gp_elements " }{TEXT -1 13 "(which can be" }}
{PARA 0 "" 0 "" {TEXT -1 77 "                      written using disjo
int cycles or lists or arrays using " }{TEXT 299 16 "array_to_disjcyc
" }{TEXT -1 2 ", " }{TEXT 300 15 "list_to_disjcyc" }{TEXT -1 2 ", " }
{TEXT 301 13 "array_to_list" }{TEXT -1 7 ", ...)," }}{PARA 0 "" 0 "" 
{TEXT -1 68 "                  2. compute the group table of any permg
roup using " }{TEXT 286 8 "gp_table" }{TEXT -1 4 " or " }{TEXT 287 12 
"abs_gp_table" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 84 "        \+
          3. compute the number of elements of order k in a permgroup \+
using " }{TEXT 290 14 "elements_order" }{TEXT -1 1 "," }}{PARA 0 "" 0 
"" {TEXT -1 65 "                  4. elements conjugate to a given ele
ment using " }{TEXT 288 15 "conjugacy_class" }{TEXT -1 74 ",\n        \+
          5. all conjugacy classes of a permutation group using " }
{TEXT 289 17 "conjugacy_classes" }{TEXT -1 207 ",\n                  6
. compute the value (i)p of a permutation as a function\n\n           \+
                     p:\{1,2,...,n\}->\{1,2,...,n\}\n\n               \+
       on a element i where p acts on the right using " }{TEXT 291 10 
"perm_value" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 66 "         \+
         7. the \"swapping number\" of a permutation using " }{TEXT 
292 4 "swap" }{TEXT -1 57 ",\n                  8. the sign of the per
mutation using " }{TEXT 293 9 "perm_sign" }{TEXT -1 1 "," }}{PARA 0 "
" 0 "" {TEXT -1 127 "                  9. the permutation matrix assoc
iated to a permutation\n                      in disjoint cycle notati
on using " }{TEXT 294 11 "perm_matrix" }{TEXT -1 21 "  (not to be conf
used" }}{PARA 0 "" 0 "" {TEXT -1 27 "                      with " }
{TEXT 298 14 "permcol_matrix" }{TEXT -1 41 " which permutes the column
s of a matrix)," }}{PARA 0 "" 0 "" {TEXT -1 251 "               10. th
e left *and* right permutation action of a \n                     perm
utation (in disjoint cycle notation) on a \n                     vecto
r (in list notation); there are two routines,\n                     on
e for the left action (" }{TEXT 296 19 "permute_vector_left" }{TEXT 
-1 27 "), one for the right action" }}{PARA 0 "" 0 "" {TEXT -1 22 "   \+
                  (" }{TEXT 297 20 "permute_vector_right" }{TEXT -1 2 
")," }}{PARA 0 "" 0 "" {TEXT -1 125 "               11. multiply a lis
t of elements of a permgroup,\n                     written in disjoin
t cycle notation using " }{TEXT 295 9 "mult_list" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 62 "               12. representation theoret
ic commands, such as " }{TEXT 278 6 "induce" }{TEXT -1 10 ", (Schur) \+
" }{TEXT 279 9 "innerprod" }{TEXT -1 27 " \n                     and \+
" }{TEXT 280 12 "is_reducible" }{TEXT -1 21 ",\n               13. " }
{TEXT 281 16 "right_coset_reps" }{TEXT -1 2 " (" }{TEXT 282 15 "left_c
oset_reps" }{TEXT -1 20 ", which agrees with " }{TEXT 283 6 "cosets" }
{TEXT -1 8 " in the " }{TEXT 284 5 "group" }{TEXT -1 10 " package)." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "You need
 to load the two packages" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "restart;with(group);with(combinat);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#7:%)DerivedSG%$LCSG%.NormalClosureG%,RandElementG%&SylowG%-areco
njugateG%'centerG%,centralizerG%%coreG%'cosetsG%'cosrepG%(derivedG%,gr
oupmemberG%+grouporderG%&interG%(invpermG%*isabelianG%)isnormalG%+issu
bgroupG%)mulpermsG%+normalizerG%&orbitG%(permrepG%%presG" }}{PARA 7 "
" 1 "" {TEXT -1 31 "Warning, new definition for Chi" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#7B%$ChiG%%bellG%)binomialG%)cartprodG%*characterG%'ch
ooseG%,compositionG%)conjpartG%+decodepartG%+encodepartG%*fibonacciG%*
firstpartG%)graycodeG%)inttovecG%)lastpartG%,multinomialG%)nextpartG%)
numbcombG%)numbcompG%)numbpartG%)numbpermG%*partitionG%(permuteG%)powe
rsetG%)prevpartG%)randcombG%)randpartG%)randpermG%*stirling1G%*stirlin
g2G%(subsetsG%)vectointG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
103 "#read(`c:/oldd/maplev4/share/algebra/group/group2_v5.mpl`);\nread
(`d:/maplestuff/group/group21_v5.mpl`);" }}{PARA 0 "" 0 "" {TEXT -1 
39 "In MAPLE V4, use group2_v4.mpl instead." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%C~~~Basic~group~theory~commands:~~~G" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%gogp_elements,~elements_order,~conjugation,~conjuga
cy_class,~conjugacy_classesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%_pcon
jugacy_classes_reps,~gp_table,~abs_gp_table,~mulperm_power,~swap,~perm
col_matrix,G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%goperm_value,~perm_si
gn,~permute_vector_left,~permute_vector_right,~mult_list,G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%[pmult_sets,~is_subset,~is_sublist,~left_co
set_reps,~right_coset_reps,~stabilizer,G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%]pcartesian_product,~cartesian_power,~conj_type,~conj_to_part,~
part_to_conj,~induce,G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%foinnerprod
,~is_reducible,~array_to_disjcyc,~list_to_disjcyc,~array_to_list,~G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^olist_to_array~...~loaded.~group2_v
5.mpl~MAPLEV5,~last~updated~12-99G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 51 "The elements
 and group table of a permutation group" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 51 " A permutation group in MAPLE is entered using the " }
{TEXT 260 9 "permgroup" }{TEXT -1 22 " command in the share " }{TEXT 
261 5 "group" }{TEXT -1 329 " package. The permutations in MAPLE are w
ritten as lists of lists. For example, the cyclic permutation (1,2,3) \+
is written [[1,2,3]] and the disjoint product (1,2)*(3,4,5) is written
 [[1,2],[3,4,5]]. The identity element 1 is written []. The subgroup o
f S_5 generated by (1,2,3) and (1,2)*(3,4,5) is given in MAPLE by the \+
command " }{TEXT 262 41 "permgroup(5,\{[[1,2,3]],[[1,2],[3,4,5]]\});" 
}{TEXT -1 66 " . This is how we shall enter groups into MAPLE in this \+
worksheet." }}{PARA 0 "" 0 "" {TEXT -1 6 "  The " }{TEXT 263 14 "group
21_v5.mpl" }{TEXT -1 9 " program " }{TEXT 259 11 "gp_elements" }{TEXT 
-1 114 " lists the elements of a group. For example, the elements of t
he symmetric groups S3,S4 and the dihedral group D4:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 46 "S3:=gp_elements(3,\{[[1,2]],[[2,3]]\});no
ps(S3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S3G7(7\"7#7$\"\"#\"\"$7#
7$\"\"\"F)7#7%F-F)F*7#7%F-F*F)7#7$F-F*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "S_3:=permgro
up(3,\{[[1,2]],[[2,3]]\});\ngroupmember([[1,2,3]],S_3);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$S_3G-%*permgroupG6$\"\"$<$7#7$\"\"#F(7#7$\"\"
\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "S4:=gp_elements(4,\{[[1,2]],[[2,3]],[[3,4]]\})
;nops(S4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#S4G7:7\"7#7$\"\"$\"\"
%7#7$\"\"#F)7#7%F-F)F*7#7%F-F*F)7#7$F-F*7#7$\"\"\"F-7$F5F(7#7%F6F-F)7#
7&F6F-F)F*7#7&F6F-F*F)7#7%F6F-F*7#7%F6F)F-7#7&F6F)F*F-7#7$F6F)7#7%F6F)
F*7$FEF37#7&F6F)F-F*7#7&F6F*F)F-7#7%F6F*F-7#7%F6F*F)7#7$F6F*7#7&F6F*F-
F)7$FRF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 56 "D4:=gp_elements(4,\{[[1,2],[3,4]],[[1,2,3,4]]
\});nops(D4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#D4G7*7\"7#7$\"\"#
\"\"%7$7$\"\"\"F)7$\"\"$F*7#7&F-F)F/F*7#7$F-F/7$F3F(7#7&F-F*F/F)7$7$F-
F*7$F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 87 "The group table of a permgroup, as a group of perm
utations, can be displayed using the " }{TEXT 265 8 "gp_table" }{TEXT 
-1 54 " command. It can be messy, as the example of D4 shows:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "gp_table(4,\{[[1,2],[3,4]],[
[1,2,3,4]]\});" }}{PARA 11 "" 1 "" {XPPMATH 303 "6#-%'matrixG6#7*7*7\"
7#7$\"\"#\"\"%7$7$\"\"\"F+7$\"\"$F,7#7&F/F+F1F,7#7$F/F17$F5F*7#7&F/F,F
1F+7$7$F/F,7$F+F17*F)F(F2F-F6F4F9F77*F-F7F(F4F2F9F)F67*F2F9F)F6F-F7F(F
47*F4F6F7F9F(F)F-F27*F6F4F9F7F)F(F2F-7*F7F-F4F(F9F2F6F)7*F9F2F6F)F7F-F
4F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The group table of S3 is n
ot so bad:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gp_table(3,\{
[[1,2]],[[2,3]]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7
(7\"7#7$\"\"#\"\"$7#7$\"\"\"F+7#7%F/F+F,7#7%F/F,F+7#7$F/F,7(F)F(F0F-F4
F27(F-F2F(F4F)F07(F0F4F)F2F(F-7(F2F-F4F(F0F)7(F4F0F2F)F-F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 61 "The first row of this may be compared wit
h the result of the " }{TEXT 264 11 "gp_elements" }{TEXT -1 9 " comman
d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "linalg[row](gp_table(
3,\{[[1,2]],[[2,3]]\}),1);\ngp_elements(3,\{[[1,2]],[[2,3]]\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7(7\"7#7$\"\"#\"\"$7#7$\"
\"\"F*7#7%F.F*F+7#7%F.F+F*7#7$F.F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7(7\"7#7$\"\"#\"\"$7#7$\"\"\"F'7#7%F+F'F(7#7%F+F(F'7#7$F+F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "A more compact looking group table
 for a permgroup can be displayed using " }{TEXT 268 12 "abs_gp_table
" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "abs_gp_
table(3,\{[[1,2]],[[2,3]]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'ma
trixG6#7)7)%\"*G&%\"aG6#\"\"\"&F*6#\"\"#&F*6#\"\"$&F*6#\"\"%&F*6#\"\"&
&F*6#\"\"'7)F)F)F-F0F3F6F97)F-F-F)F3F0F9F67)F0F0F6F)F9F-F37)F3F3F9F-F6
F)F07)F6F6F0F9F)F3F-7)F9F9F3F6F-F0F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6(/&%\"aG6#\"\"\"7\"/&F%6#\"\"#7#7$F,\"\"$/&F%6#F/7#7$F'F,/&F%6#\"\"%7
#7%F'F,F//&F%6#\"\"&7#7%F'F/F,/&F%6#\"\"'7#7$F'F/" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 44 "abs_gp_table(4,\{[[1,2],[3,4]],[[1,2,3,4]]\}
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7+7+%\"*G&%\"aG6#\"
\"\"&F*6#\"\"#&F*6#\"\"$&F*6#\"\"%&F*6#\"\"&&F*6#\"\"'&F*6#\"\"(&F*6#
\"\")7+F)F)F-F0F3F6F9F<F?7+F-F-F)F3F0F9F6F?F<7+F0F0F<F)F6F3F?F-F97+F3F
3F?F-F9F0F<F)F67+F6F6F9F<F?F)F-F0F37+F9F9F6F?F<F-F)F3F07+F<F<F0F6F)F?F
3F9F-7+F?F?F3F9F-F<F0F6F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"aG6#
\"\"\"7\"/&F%6#\"\"#7#7$F,\"\"%/&F%6#\"\"$7$7$F'F,7$F3F//&F%6#F/7#7&F'
F,F3F//&F%6#\"\"&7#7$F'F3/&F%6#\"\"'7$FAF./&F%6#\"\"(7#7&F'F/F3F,/&F%6
#\"\")7$7$F'F/7$F,F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 40 "Conjugacy classes of a permutation group
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The program " }{TEXT 266 15 "c
onjugacy_class" }{TEXT -1 126 " lists the conjugacy class of an elemen
t of a group. For example, the conjugacy class of [[1,2]] in the symme
tric group S3 is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "conjug
acy_class([[1,2]],3,\{[[1,2]],[[2,3]]\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<%7#7$\"\"\"\"\"$7#7$\"\"#F'7#7$F&F*" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 61 "The conjugacy class of  [[2,4]] in the symmetri
c group D4 is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "conjugacy
_class([[2,4]],4,\{[[1,2],[3,4]],[[1,2,3,4]]\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$7#7$\"\"\"\"\"$7#7$\"\"#\"\"%" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 12 "The program " }{TEXT 267 17 "conjugacy_classes" }
{TEXT -1 102 " lists all the conjugacy classes of a group. For example
, the conjugacy classes of S3, S4, and D4 are:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 47 "conjugacy_classes(4,\{[[1,2]],[[2,3]],[[3,4]]
\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<'<%7$7$\"\"\"\"\"%7$\"\"#\"\"
$7$7$F'F+7$F*F(7$7$F'F*7$F+F(<(7#7&F'F*F+F(7#7&F'F(F+F*7#7&F'F(F*F+7#7
&F'F+F(F*7#7&F'F+F*F(7#7&F'F*F(F+<#7\"<*7#7%F'F+F*7#7%F'F*F+7#7%F'F(F*
7#7%F'F(F+7#7%F'F+F(7#7%F'F*F(7#7%F*F+F(7#7%F*F(F+<(7#F-7#F)7#F07#F&7#
F.7#F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "conjugacy_classes
(3,\{[[1,2]],[[2,3]]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%<%7#7$\"
\"\"\"\"$7#7$\"\"#F(7#7$F'F+<#7\"<$7#7%F'F(F+7#7%F'F+F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "conjugacy_classes(4,\{[[1,2],[3,4]]
,[[1,2,3,4]]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'<#7$7$\"\"\"\"\"
$7$\"\"#\"\"%<#7\"<$7$7$F'F+7$F*F(7$7$F'F*7$F(F+<$7#F&7#F)<$7#7&F'F*F(
F+7#7&F'F+F(F*" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT -1 44 "Right and left cosets of a permutation group" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "G:=Symm(4,4); H:=Symm(3,4);
\n" }{TEXT -1 88 "G is the symmetric group S4 and H is S3 regarded as \+
the \n\nsubgroup which leaves 4 fixed." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"GG-%*permgroupG6$\"\"%<$7#7&\"\"\"\"\"#\"\"$F(7#7$F,F-" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG-%*permgroupG6$\"\"%<$7#7$\"\"\"
\"\"#7#7%F,F-\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "s4:=g
p_elements(op(1,G),op(2,G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s4G
-%)elementsG6$\"\"%<$7#7&\"\"\"\"\"#\"\"$F(7#7$F,F-" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "s3:=gp_elements(op(1,H),op(2,H));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s3G7(7\"7#7$\"\"#\"\"$7#7$\"\"\"F)7
#7%F-F)F*7#7%F-F*F)7#7$F-F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "To
 obtain a set of coset representatives of S3 in S4, type" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "C1:=right_coset_reps(G,H);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G<&7\"7#7&\"\"\"\"\"%\"\"$\"\"#7#
7%F,F*F+7#7$F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 144 "mult_sets(\{C1[1]\},convert(s3,set));\nmult_sets(
\{C1[2]\},convert(s3,set));\nmult_sets(\{C1[3]\},convert(s3,set));\nmu
lt_sets(\{C1[4]\},convert(s3,set));\n" }{TEXT -1 76 "This checks that \+
C1 is indeed a complete set of right coset representatives." }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<(7\"7#7$\"\"\"\"\"$7#7%F'F(\"\"#7#7$F+F(7#7
$F'F+7#7%F'F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(7$7$\"\"\"\"\"%7$
\"\"#\"\"$7#7&F&F'F*F)7#7%F&F'F)7#7%F&F'F*7#F%7#7&F&F'F)F*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<(7$7$\"\"\"\"\"$7$\"\"#\"\"%7#7&F&F'F)F*7#7
&F&F)F*F'7#7%F&F)F*7#7%F)F*F'7#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<
(7#7&\"\"\"\"\"#\"\"$\"\"%7#7&F&F(F)F'7#7%F&F(F)7#7%F'F(F)7$7$F&F'7$F(
F)7#F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "To obtain a set of left
 coset representatives of S3 in S4, type (coset(G,H); or)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "C2:=left_coset_reps(G,H);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#C2G<&7\"7#7&\"\"\"\"\"#\"\"$\"\"%7#7%F*F+
F,7#7$F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "mult_sets(convert(s3,set),\{C2[1]
\});\nmult_sets(convert(s3,set),\{C2[2]\});\nmult_sets(convert(s3,set)
,\{C2[3]\});\nmult_sets(convert(s3,set),\{C2[4]\});\n" }{TEXT -1 76 "T
his checks that C2 is indeed a complete set of leftt coset representat
ives." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(7\"7#7$\"\"\"\"\"$7#7%F'F(
\"\"#7#7$F+F(7#7$F'F+7#7%F'F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(7
#7&\"\"\"\"\"#\"\"$\"\"%7$7$F&F)7$F'F(7#F+7#7%F&F(F)7#7&F&F(F'F)7#7%F&
F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(7#7%\"\"\"\"\"%\"\"#7#7&F&F'
F(\"\"$7#7&F&F+F'F(7$7$F&F+7$F(F'7#7%F(F+F'7#F0" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<(7#7&\"\"\"\"\"%\"\"$\"\"#7#7%F&F'F(7#7&F&F)F'F(7#7%F)
F'F(7$7$F&F)7$F(F'7#F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 48 "Elements of a given order of a permutatio
n group" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 270 13 "mul
perm_power" }{TEXT -1 52 " command returns the k-th power of a disjoin
t cycle:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "mulperm_power([
[1,2,3]],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7%\"\"\"\"\"$\"\"#" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The elements of a given order c
an be determined using the " }{TEXT 269 14 "elements_order" }{TEXT -1 
22 " command. For example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "elements_order(3,\{[[1,2]],[[2,3]]\},2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$\"\"%7&7\"7#7$\"\"#\"\"$7#7$\"\"\"F)7#7$F-F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "elements_order(3,\{[[1,2]],[
[2,3]]\},3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"$7%7\"7#7%\"\"\"
\"\"#F$7#7%F)F$F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 60 "The action of the permutation group G on \{1,2,
...,degree(G)\}" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 
271 10 "perm_value" }{TEXT -1 70 " command determines the value (i)p o
f a permutation p at an element i:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 125 "P1:=[[1,2,3]]; \nP2:=[[1,2],[3,4]];\nperm_value(P1,2
);\nperm_value(P1,4); \nperm_value(P1,3);\nperm_value(P2,2);\nperm_val
ue(P2,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G7#7%\"\"\"\"\"#\"\"
$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P2G7$7$\"\"\"\"\"#7$\"\"$\"\"%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 272 9 
"swap(p,n)" }{TEXT -1 155 " command determines the swapping number of \+
p in S_n, where the swapping number of a permutation is the number of \+
inequalities i<j, 1<=i,j<=n, swapped by p." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 "swap(P1,3);\nswap(P2,4);\nswap([[1,2]],5);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 4 "The " }{TEXT 273 9 "perm_sign" }{TEXT -1 57 " comm
and determines the sign of the permutation p in S_n." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 56 "perm_sign(P1,3); \nperm_sign(P2,4);\nperm
_sign([[1,2]],5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 274 11 "perm_
matrix" }{TEXT -1 112 " command computes the associated permutation ma
trix of a permutation of \{1,2,...,n\} in disjoint cycle notation: " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "L:=[[1,2,4],[3,6]];\nperm_
matrix(L);\nperm_matrix(L,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"L
G7$7%\"\"\"\"\"#\"\"%7$\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'matrixG6#7(7(\"\"!\"\"\"F(F(F(F(7(F(F(F(F)F(F(7(F(F(F(F(F(F)7(F)F(F(
F(F(F(7(F(F(F(F(F)F(7(F(F(F)F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'matrixG6#7)7)\"\"!\"\"\"F(F(F(F(F(7)F(F(F(F)F(F(F(7)F(F(F(F(F(F)F(7
)F)F(F(F(F(F(F(7)F(F(F(F(F)F(F(7)F(F(F)F(F(F(F(7)F(F(F(F(F(F(F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 275 19 "permute_vector
_left" }{TEXT -1 5 " and " }{TEXT 276 20 "permute_vector_right" }
{TEXT -1 106 " commands will allow you to compute the permutation acti
on of a permgroup on a vector (written as a list):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 196 "g1:=[[1,4,3,2]]; g2:=[[9,10,11,12]];\nw:=[
1,2,3,4,5,6,7,8,9,10,11,12];\ng3:=invperm(g1);\npermute_vector_left(g1
,w);\npermute_vector_left(g2,w);\npermute_vector_right(g1,w);\npermute
_vector_right(g3,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1G7#7&\"\"
\"\"\"%\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2G7#7&\"\"*\"
#5\"#6\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG7.\"\"\"\"\"#\"\"$
\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#g3G7#7&\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7.\"\"#\"\"$\"\"%\"\"\"\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#
6\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"\"\"\"\"#\"\"$\"\"%\"\"&
\"\"'\"\"(\"\")\"#7\"\"*\"#5\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.
\"\"%\"\"\"\"\"#\"\"$\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7.\"\"#\"\"$\"\"%\"\"\"\"\"&\"\"'\"\"(\"\")
\"\"*\"#5\"#6\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 
277 9 "mult_list" }{TEXT -1 61 " command will multiply together a list
 of permgroup elements:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "
r:=[[1,2,3,4,5,6,7,8,9,10,11,12]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"rG7#7.\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "for j from 1 to 11 do r.j
:=mult_list([seq(r,i=1..j)]); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#r1G7#7#7.\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G7$7(\"\"\"\"\"$\"\"&\"\"(\"
\"*\"#67(\"\"#\"\"%\"\"'\"\")\"#5\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#r3G7%7&\"\"\"\"\"%\"\"(\"#57&\"\"#\"\"&\"\")\"#67&\"\"$\"\"'\"\"
*\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r4G7&7%\"\"\"\"\"&\"\"*7%
\"\"#\"\"'\"#57%\"\"$\"\"(\"#67%\"\"%\"\")\"#7" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#r5G7#7.\"\"\"\"\"'\"#6\"\"%\"\"*\"\"#\"\"(\"#7\"\"&
\"#5\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r6G7(7$\"\"\"\"\"
(7$\"\"#\"\")7$\"\"$\"\"*7$\"\"%\"#57$\"\"&\"#67$\"\"'\"#7" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#r7G7#7.\"\"\"\"\")\"\"$\"#5\"\"&\"#7\"\"(
\"\"#\"\"*\"\"%\"#6\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r8G7&7%
\"\"\"\"\"*\"\"&7%\"\"#\"#5\"\"'7%\"\"$\"#6\"\"(7%\"\"%\"#7\"\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r9G7%7&\"\"\"\"#5\"\"(\"\"%7&\"\"#
\"#6\"\")\"\"&7&\"\"$\"#7\"\"*\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$r10G7$7(\"\"\"\"#6\"\"*\"\"(\"\"&\"\"$7(\"\"#\"#7\"#5\"\")\"\"'\"
\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r11G7#7.\"\"\"\"#7\"#6\"#5\"
\"*\"\")\"\"(\"\"'\"\"&\"\"%\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "mult_list([[[1,2],[3,5]],[[1,2]]]);\n" }{TEXT -1 34 "
multiplies (1,2)(3,5) times (1,2)." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7#7$\"\"$\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "We can take a \+
permutation, say written as a list 32415 (so 1 is sent to 4, 2 is left
 alone, ...), convert it to a disjoint cycle, and then permute the cor
responding columns of a martix):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 117 "perm:=[3,2,4,1,5];\np:=list_to_disjcyc(perm);\norigi
nal:=diag(1,2,3,4,5);\npermuted_by_perm:=permcol_matrix(original,p);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%permG7'\"\"$\"\"#\"\"%\"\"\"\"\"&
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG7#7%\"\"\"\"\"%\"\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)originalG-%'matrixG6#7'7'\"\"\"\"\"
!F+F+F+7'F+\"\"#F+F+F+7'F+F+\"\"$F+F+7'F+F+F+\"\"%F+7'F+F+F+F+\"\"&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1permuted_by_permG-%'matrixG6#7'7'
\"\"!F*F*\"\"\"F*7'F*\"\"#F*F*F*7'\"\"$F*F*F*F*7'F*F*\"\"%F*F*7'F*F*F*
F*\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 33 "Representation-theoretic commands" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "f:=a->Chi([1,2],conj_type(a,3)):\nf([[2,1]]);\n
" }{TEXT -1 43 "This is the character of the standard repn." }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "is_reducible(Symm(3,3),f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%KIf~f~is~a~character~then~it~is~irreducibleG" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 4 "The " }{TEXT 304 13 "inner product" }{TEXT -1 118 " of \+
two functions f1, f2 on a group G is given by |G|^(-1) times the sum o
f f1(x)*conjugate(f2(x)), as x runs over G. " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "innerprod(Symm(3,3),f,f);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "We indu
ce f from S3 to S4 and compute a few values (f2(1,3,2)=f2(1,4,2) since
 f2 is a class function). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "f2:=g->induce(G,H,f,g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2GR
6#%\"gG6\"6$%)operatorG%&arrowGF(-%'induceG6&%\"GG%\"HG%\"fG9$F(F(F(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "g:=[[1,4,2]]:\nprint(`ma
trix element: `,g,`    character value at this element: `,f2(g));\ng:=
[[1,3,2]]:\nprint(`matrix element: `,g,`    character value at this el
ement: `,f2(g));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%1matrix~element:~
G7#7%\"\"\"\"\"%\"\"#%F~~~~character~value~at~this~element:~G!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&%1matrix~element:~G7#7%\"\"\"\"\"$\"\"
#%F~~~~character~value~at~this~element:~G!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Is the
 induced character irreducible?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "is_reducible(G,f2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%IIf~f~
is~a~character~then~it~is~reducibleG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 147 "In other words, no it's not irreducible. Indeed, if the induce
d rep was irreducible then its inner product must equal 1, but we have
 the following:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "innerpro
d(G,f2,f2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 202 "Since the inner product of a repn with itself \+
is the sum of the squares of the multiplicities, this forces f2 to be \+
the sum of 3 irreducibles of S3. Which 3 repns of S3 occur in the deco
mposition of f2?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "The values of
 this induced character on the conjugacy classes of S4 can be computed
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "R:=conjugacy_classes_r
eps(op(1,G),op(2,G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG<'7\"7#
7$\"\"\"\"\"$7#7%F)F*\"\"#7#7&F)F-F*\"\"%7$7$F)F07$F-F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "seq([g,f2(g)],g=R);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6'7$7\"\"\")7$7#7$\"\"\"\"\"$\"\"!7$7#7%F)F*\"\"#!\"
\"7$7#7&F)F/F*\"\"%F+7$7$7$F)F47$F/F*F+" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 114 "This listing is not in the same order as the partitions \+
listing, so we must reorder it before comparing its values" }}{PARA 0 
"" 0 "" {TEXT -1 46 "with those given by Maple's character command:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "R1:=[R[1],R[2],R[5],R[4],R
[3]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R1G7'7\"7#7$\"\"\"\"\"#7$F
(7$\"\"$\"\"%7#7&F)F*F-F.7#7%F)F*F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "seq([g,f2(g)],g=R1);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6'7$7\"\"\")7$7#7$\"\"\"\"\"#\"\"!7$7$F(7$\"\"$\"\"%F+7$7#7&F)F*F/F0F
+7$7#7%F)F*F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "What linear
 combination of the irreducible characters of S4 gives this list of va
lues?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "character(4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"\"F(F(F(F(7'\"\"$F
(!\"\"\"\"!F+7'\"\"#F,F.F+F,7'F*F+F+F,F(7'F(F+F(F(F+" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 59 "We check which column corresponds to which conj
ugacy class:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "p4:=partiti
on(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G7'7&\"\"\"F'F'F'7%F'F'
\"\"#7$F)F)7$F'\"\"$7#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "for i from nops(p4) to 1 by -1 do\n seq(Chi(p4[i],p),p=p4);\nod;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"F#F#F#F#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"$\"\"\"!\"\"\"\"!F%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6'\"\"#\"\"!F#!\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$!\"
\"F$\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"!\"\"F#F#F$" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 228 "Thus, for example, the 2nd col
umn corresponds to the value of the character on the conjugacy class r
epresented by (1,2), since it is of type [1,1,2].  We compare and see \+
that the induced repn is the sum of the middle three rows:" }}{PARA 0 
"" 0 "" {TEXT -1 61 "                       f2 = Chi(p4[2])+Chi(p4[3])
+Chi(p4[4])." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "seq([part_t
o_conj(p),Chi(p4[2],p)+Chi(p4[3],p)+Chi(p4[4],p)],p=p4);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6'7$7&7#\"\"\"7#\"\"#7#\"\"$7#\"\"%\"\")7$7%F%F'7$
F*F,\"\"!7$7$7$F&F(F0F17$7$F%7%F(F*F,!\"\"7$7#7&F&F(F*F,F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "seq([g,f2(g)],g=R1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6'7$7\"\"\")7$7#7$\"\"\"\"\"#\"\"!7$7$F(7$\"\"$\"
\"%F+7$7#7&F)F*F/F0F+7$7#7%F)F*F/!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 79 "This is consistent with the fact that the inner product o
f f2 with itself is 3." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 26 "An electronic copy of the " }{TEXT 257 14 "group21_v5.mpl" }
{TEXT -1 15 " file and this " }{TEXT 258 12 "symm_gp0.mws" }{TEXT -1 
26 " file can be obtained from" }}{PARA 0 "" 0 "" {TEXT -1 39 "http://
web.usna.navy.mil/~wdj/group.htm" }}{PARA 0 "" 0 "" {TEXT -1 10 "An up
date " }{TEXT 302 12 "group_v6.mpl" }{TEXT -1 47 " for the upcoming re
lease of MAPLE6 is planned." }}}}{MARK "11 2 2" 46 }{VIEWOPTS 1 1 0 1 
1 1803 }