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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 57 " Beispiele zur Benutzung v
on 'groups' in Maple und zu \2475 " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 23 " (a) zum Beispiel 1.5.3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "
Es geht um die Darstellung der Quaternionengruppe durch Relationen f
\374r " }{TEXT 257 4 "drei" }{TEXT -1 9 " Erzeuger" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(group); \+
" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and
 trace have been redefined and unprotected\n" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7>%)DerivedSG%$LCSG%.NormalClosureG%,RandElementG%-SnCo
njugatesG%&SylowG%-areconjugateG%'centerG%,centralizerG%%coreG%'cosets
G%'cosrepG%(derivedG%)elementsG%,groupmemberG%+grouporderG%&interG%(in
vpermG%*isabelianG%)isnormalG%+issubgroupG%)mulpermsG%+normalizerG%&or
bitG%'parityG%(permrepG%%presG%+transgroupG" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 78 "A:=matrix(2,2,[0,i,i,0]);B:=matrix(2,2,[0,1,-1,0])
;C:=matrix(2,2,[-1,0,0,-1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG
K%'matrixG6#7$7$\"\"!%\"iG7$F+F*Q)pprint166\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BGK%'matrixG6#7$7$\"\"!\"\"\"7$!\"\"F*Q)pprint176\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CGK%'matrixG6#7$7$!\"\"\"\"!7$
F+F*Q)pprint186\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Die Quaterni
onengruppe wird von A,B erzeugt und enth\344lt dann " }{XPPEDIT 18 0 "
C=A^2" "6#/%\"CG*$%\"AG\"\"#" }{TEXT -1 11 ",  (ebenso " }{XPPEDIT 18 
0 "C=B^2" "6#/%\"CG*$%\"BG\"\"#" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" 
{TEXT -1 68 "Dann wird Q auch von \{A,B,C\} erzeugt. Es gelten folgend
e Relationen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "G:=grelgro
up(\{x,y,z\}, \{[z,z],[x,z,1/x,1/z],[y,z,1/y,1/z],[x,x,1/z],[y,y,1/z],
[x,y,1/x,1/y,1/z]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%*grel
groupG6$<%%\"xG%\"yG%\"zG<(7$F+F+7&F)F+*&\"\"\"F0F)!\"\"*&F0F0F+F17&F*
F+*&F0F0F*F1F27%F)F)F27%F*F*F27'F)F*F/F4F2" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "grouporder(G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Die elemente von G kann \+
man sich so anzeigen lassen ([ ] steht dabie fuer das neutrale Element
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "U:=subgrel(\{x=[]\},G
):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "cosets(U);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#<*7#%\"zG7$%\"xGF%7#F'7#%\"yG7$F*F%7$F'F*7\"7%F'F%F*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Bemerkung: " }{TEXT 258 4 "We
nn" }{TEXT -1 281 " Maple richtig gerechnet hat, dann ist G bis auf Is
omorphie die freie von    \{x,y,z\}    erzeugte Gruppe mit den angegeb
enen Relationen! Da in der Quaternionengruppe die gleichen Relationen \+
gelten und Q ebenfalls aus 8 Elementen besteht, sind zwangsl\344ufig G
 und Q isomorphe Gruppen." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 " (b) zu Aufgabe (11)" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Jetzt geht es um die Darstellung \+
der Quaternionengruppe als freie von    \{x,y\}   erzeugte Gruppe mit \+
geeigneten Relationen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "with(group); " }}{PARA 7 "" 1 "" {TEXT -1 
80 "Warning, the protected names norm and trace have been redefined an
d unprotected\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7>%)DerivedSG%$LCSG
%.NormalClosureG%,RandElementG%-SnConjugatesG%&SylowG%-areconjugateG%'
centerG%,centralizerG%%coreG%'cosetsG%'cosrepG%(derivedG%)elementsG%,g
roupmemberG%+grouporderG%&interG%(invpermG%*isabelianG%)isnormalG%+iss
ubgroupG%)mulpermsG%+normalizerG%&orbitG%'parityG%(permrepG%%presG%+tr
ansgroupG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Die Quaternionengrup
pe wird von A,B, erzeugt , wobei" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 
"A:=matrix(2,2,[0,i,i,0]);B:=matrix(2,2,[0,1,-1,0]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"AGK%'matrixG6#7$7$\"\"!%\"iG7$F+F*Q)pprint196\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGK%'matrixG6#7$7$\"\"!\"\"\"7
$!\"\"F*Q)pprint206\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Die Quat
ernionengruppe wird von A,B, erzeugt mit den Relationen" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "G:=grelgroup(\{x,y\}, \{[x,x,x,x],[
y,y,y,y],[x,x,1/y,1/y],[x,y,y,1/x,1/y,1/y],[x,y,1/x,1/y,1/y,1/y]\});" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%*grelgroupG6$<$%\"xG%\"yG<'7
&F)F)F)F)7&F*F*F*F*7&F)F)*&\"\"\"F0F*!\"\"F/7(F)F*F**&F0F0F)F1F/F/7(F)
F*F3F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "grouporder(G)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 24 "Die Elemente von G sind:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "V:=subgrel(\{x=[]\},G):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "cosets(V);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*7$%\"
xGF%7&F%F%F%%\"yG7%F'F'F'7$F%F'7\"7%F%F%F%7#F'7#F%" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 40 "Bemerkung analog zu der am Ende von (a)." }}}}
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