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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 66 "Abstract Algebra with Map
le\nChapter 4: Finite Groups and Subgroups" }}{PARA 19 "" 0 "" {TEXT 
-1 3 "by " }{URLLINK 17 "Alec Mihailovs" 4 "http://webpages.shepherd.e
du/amihailo" "" }{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 
"4: Finite Groups and Subgroups" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
18 "Orders of Elements" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "After l
oading the number theory package (we did it in the previous section), \+
orders of elements of groups " }{XPPEDIT 18 0 "U(n);" "6#-%\"UG6#%\"nG
" }{TEXT -1 29 " can be evaluated as follows:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "order(7,15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "order(31,42);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 23 "The cyclic subgroup of " }{XPPEDIT 18 0 "U(n);" "6#-%\"UG
6#%\"nG" }{TEXT -1 14 " generated by " }{TEXT 256 1 "a" }{TEXT -1 45 "
, can be found using the following procedure:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "cycleU:=(c,n)->[seq(c^(i-1) mod n, i=1..order(c,
n))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "cycleU(7,15);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"\"\"(\"\"%\"#8" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "cycleU(31,42);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7(\"\"\"\"#J\"#P\"#8\"#D\"#>" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 56 "In general, the cyclic subgroup generated by an element
 " }{TEXT 258 1 "c" }{TEXT -1 24 " of an (extended) group " }{TEXT 
257 1 "g" }{TEXT -1 44 " can be found using the following procedure:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "Cycle:=proc(c,g) local v
,a;\nv:=[g[3]]; a:=c; \nwhile not a=g[3] do v:=[op(v),a]; a:=g[2](a,c)
 od; v end: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "For example, the \+
cyclic subgroup generated by 10 in " }{XPPEDIT 18 0 "Z[25];" "6#&%\"ZG
6#\"#D" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "C
ycle(10,Z(25));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!\"#5\"#?\"\"
&\"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Another example, " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "map(matrix,Cycle([[1,2],[3,4
]],GL2(5)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7*-%'matrixG6#7$7$\"\"
\"\"\"!7$F*F)-F%6#7$7$F)\"\"#7$\"\"$\"\"%-F%6#7$7$F0F*7$F*F0-F%6#7$7$F
0F37$F)F2-F%6#7$7$F3F*7$F*F3-F%6#7$7$F3F27$F0F)-F%6#7$7$F2F*7$F*F2-F%6
#7$7$F2F)7$F3F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "The orders of \+
elements can be found as the orders of the cyclic subgroups generated \+
by them:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Ord:=proc(c,g) \+
local a,n;\na:=c; n:=1; \nwhile not a=g[3] do n:=n+1; a:=g[2](a,c) od;
 n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Ord([[1,2],[3,4]
],SL2(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "Ord([[2,3],[4,5]],GL2(11));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "Ord(715,Z(1001));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Center and Centralizers" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The centralizer of an element " }
{TEXT 265 2 "a " }{TEXT -1 21 "of an extended group " }{TEXT 264 1 "G
" }{TEXT -1 30 " can be determined as follows:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 137 "CentralizerE:=proc(a,G) local i,v;\nv:=[]; fo
r i to nops(G[1]) do if G[2](G[1][i],a)=G[2](a,G[1][i]) then v:=[op(v)
,G[1][i]] fi od; v end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For ex
ample, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "CentralizerE(8,C
yclic(8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*\"\"\"\"\"#\"\"$\"\"%
\"\"&\"\"'\"\"(\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Cen
tralizerE(8,Dihedral(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"
\"\"$\"\"'\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The centralize
r of a" }{TEXT 267 1 " " }{TEXT -1 7 "subset " }{TEXT 269 1 "S" }
{TEXT -1 0 "" }{TEXT 268 1 " " }{TEXT -1 21 "of an extended group " }
{TEXT 266 1 "G" }{TEXT -1 31 " can be determined as follows: " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "CentralizerS:=proc(S,G) loc
al i,j,v; \nv:=[]; for i to nops(G[1]) do j:=1; while not j=nops(S)+1 \+
and \nG[2](G[1][i],S[j])=G[2](S[j],G[1][i]) do j:=j+1 od; if j=nops(S)
+1 then v:=[op(v),G[1][i]] fi od; v end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "CentralizerS([7,8],Dihedral(4));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "T
he center is the centralizer of the group: " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 32 "Center:=G->CentralizerS(G[1],G):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "Center(Dihedral(7));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
Center(Dihedral(10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"\"\"'
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(matrix,Center(GL2(3
)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'matrixG6#7$7$\"\"\"\"\"!7
$F*F)-F%6#7$7$\"\"#F*7$F*F0" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 5 "No
te." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Why do we need two commands
 for a centralizer - " }{TEXT 279 12 "CentralizerE" }{TEXT -1 5 " and \+
" }{TEXT 280 12 "CentralizerS" }{TEXT -1 210 "? Why can't we use one c
ommand, Centralizer, for both cases? The problem is that some elements
 of a group can be equal to some sets of other elements. For example, \+
a group can contain elements 1, 2, and \{1,2\}. " }}{PARA 0 "" 0 "" 
{TEXT -1 89 "What would the Centralizer(\{1,2\}) be in that case? The \+
centralizer of the element \{1,2\}, " }{TEXT 281 12 "CentralizerE" }
{TEXT -1 49 "(\{1,2\}), or the centralizer of the subset \{1,2\}, " }
{TEXT 282 12 "CentralizerS" }{TEXT -1 138 "(\{1,2\})? Since we can't d
istinguish between these two cases by checking the type of an argument
, we need two different commands for that. " }}}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 15 "A Subgroup Test" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
89 "According to Theorem 3.3 on p. 62 of Dr. Gallian's text, to test w
hether a finite subset " }{TEXT 259 1 "H" }{TEXT -1 18 " is a subgroup
 of " }{TEXT 260 1 "G" }{TEXT -1 27 ", it is enough to check if " }
{TEXT 261 1 "H" }{TEXT -1 16 " is a subset of " }{TEXT 292 1 "G" }
{TEXT -1 47 " and it is closed under the group operation of " }{TEXT 
262 1 "G" }{TEXT -1 16 ". So we can use " }{TEXT 263 4 "isCP" }{TEXT 
-1 11 " for that: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "isSub
group:=(h,G)->verify(\{op(h)\},\{op(G[1])\},`subset`) and isCP(h,G[2])
:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For example, " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "isSubgroup(Cycle(8,Z(33)),Z(33));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "isSubgroup(Cycle([[1,2],[2,0]],GL2(5)),SL2(5));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 86 "Certainly, cyclic subgroups are subgroups :-) as well as \+
the center and centralizers: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "isSubgroup(CentralizerE(11,Dihedral(6)),Dihedral(6));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "isSubgroup(CentralizerS(\{[[1,2],[3,4]],[[1,3],[2,4]]
\},GL2(5)),GL2(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "isSubgroup(Center(SL2(4)),SL
2(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 29 "Finally, an opposite example:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 26 "isSubgroup(Z(5)[1],Z(10));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Exercises" }}{EXCHG 
{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 270 1 "1" }{TEXT -1 25 ". Find t
he order of 7 in " }{TEXT 271 1 "U" }{TEXT -1 10 "(100) and " }{TEXT 
272 1 "U" }{TEXT -1 6 "(11). " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }
{TEXT 273 1 "2" }{TEXT -1 25 ". Find the order of 6 in " }{XPPEDIT 18 
0 "Z[7];" "6#&%\"ZG6#\"\"(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Z[8];" "6
#&%\"ZG6#\"\")" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Z[9];" "6#&%\"ZG6#\"
\"*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Z[10];" "6#&%\"ZG6#\"#5" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "Z[11];" "6#&%\"ZG6#\"#6" }{TEXT -1 6 ", and \+
" }{XPPEDIT 18 0 "Z[12];" "6#&%\"ZG6#\"#7" }{TEXT -1 2 ". " }}{PARA 
14 "" 0 "" {TEXT -1 0 "" }{TEXT 274 1 "3" }{TEXT -1 30 ". Find the cyc
lic subgroup of " }{TEXT 275 1 "U" }{TEXT -1 23 "(145) generated by 19
. " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 276 1 "4" }{TEXT -1 30 ". \+
Find the cyclic subgroup of " }{TEXT 277 2 "GL" }{TEXT -1 4 "(2, " }
{XPPEDIT 18 0 "Z[6];" "6#&%\"ZG6#\"\"'" }{TEXT -1 15 ") generated by \+
" }{XPPEDIT 18 0 "matrix([[2, 5], [1, 2]]);" "6#-%'matrixG6#7$7$\"\"#
\"\"&7$\"\"\"F(" }{TEXT -1 3 " . " }}{PARA 14 "" 0 "" {TEXT -1 1 " " }
{TEXT 278 1 "5" }{TEXT -1 40 ". Find the order of cyclic subgroups of \+
" }{TEXT 283 2 "SL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[11];" "6#&%\"
ZG6#\"#6" }{TEXT -1 6 ") and " }{TEXT 285 2 "SL" }{TEXT -1 4 "(2, " }
{XPPEDIT 18 0 "Z[12];" "6#&%\"ZG6#\"#7" }{TEXT -1 15 ") generated by \+
" }{XPPEDIT 18 0 "matrix([[1, 2], [3, 7]]);" "6#-%'matrixG6#7$7$\"\"\"
\"\"#7$\"\"$\"\"(" }{TEXT -1 3 " . " }}{PARA 14 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 284 1 "6" }{TEXT -1 50 ". Find the centralizer of 3 in the di
hedral group " }{XPPEDIT 18 0 "D[6];" "6#&%\"DG6#\"\"'" }{TEXT -1 2 ".
 " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 286 1 "7" }{TEXT -1 28 ". F
ind the centralizer of \{ " }{XPPEDIT 18 0 "matrix([[1, 3], [1, 2]]);
" "6#-%'matrixG6#7$7$\"\"\"\"\"$7$F(\"\"#" }{TEXT -1 3 " , " }
{XPPEDIT 18 0 "matrix([[1, 2], [3, 4]]);" "6#-%'matrixG6#7$7$\"\"\"\"
\"#7$\"\"$\"\"%" }{TEXT -1 6 " \} in " }{TEXT 287 2 "GL" }{TEXT -1 4 "
(2, " }{XPPEDIT 18 0 "Z[5];" "6#&%\"ZG6#\"\"&" }{TEXT -1 3 "). " }}
{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 288 1 "8" }{TEXT -1 21 ". Find t
he center of " }{TEXT 289 2 "SL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[
6];" "6#&%\"ZG6#\"\"'" }{TEXT -1 3 "). " }}{PARA 14 "" 0 "" {TEXT -1 
0 "" }{TEXT 290 1 "9" }{TEXT -1 18 ". Test if the set " }{XPPEDIT 18 
0 "\{1, 4, 11, 14, 16, 19, 26, 29, 31, 34, 41, 44\};" "6#<.\"\"\"\"\"%
\"#6\"#9\"#;\"#>\"#E\"#H\"#J\"#M\"#T\"#W" }{TEXT -1 18 " is a subgroup
 of " }{TEXT 291 1 "U" }{TEXT -1 5 "(45)." }}}}}}{MARK "0 0 0" 0 }
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