Tirohia te wāhanga i raro nei, kātahi ka whakautu i te pātai i muri mai i te reo kotahi me te wāhanga:  Passage: Ko te ariā o nāianei o tētahi rōpū ātaahua i whakawhanakehia mai i ētahi āpure pāngarau. Ko te hihiri taketake mo te ariā rōpū ko te rapu rongoā o ngā whārite taumaha o te waeine nui ake i te 4. Ko te kaikaikarau Wīwī o te rau tau 19, ko Évariste Galois, e whakawhānui ana i te mahi o mua o Paolo Ruffini me Joseph-Louis Lagrange, i hoatu he paearu mo te whakaoti i tētahi whārite taumaha motuhake i runga i te rōpū taurite o ōna pakiaka (ngā rongoā). Ko ngā huānga o taua rōpū Galois e ōrite ana ki ētahi whakarerekētanga o ngā pakiaka. I te tuatahi, i whakakahoretia nga whakaaro o Galois e ona hoa, a i whakaputaina noa i muri i te matenga. I tirotirohia ngā rōpū whakarerekētanga whānui ake e Augustin Louis Cauchy. Ko Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) e whakarato ana i te tautuhinga tuatahi o te roopu mutunga. Ko te Geometry te waahanga tuarua i whakamahia ai nga roopu, ina koa ko nga roopu taurite hei waahanga o te kaupapa Erlangen 1872 a Felix Klein. I muri i te putanga mai o nga āhuahanga hou pēnei i te āhuahanga hyperbolic me te āhuahanga whakaata, i whakamahia e Klein te ariā rōpū hei whakarite i a raatau i roto i te huarahi tika ake. I te haere whakamua o enei whakaaro, i whakatuhia e Sophus Lie te rangahau o nga roopu Lie i te 1884. Ko te mara tuatoru e whai hua ana ki te ariā roopu ko te ariā tau. I whakamahia āta ētahi hanganga rōpū Abelian i roto i te mahi tau-ariā a Carl Friedrich Gauss's Disquisitiones Arithmeticae (1798), ā, i tino mārama ake e Leopold Kronecker. I te tau 1847, i whakamātau wawe a Ernst Kummer ki te whakamatau i te Kaupapa Whakamutunga a Fermat mā te whakawhanake i ngā rōpū e whakaahua ana i te whakareatanga ki ngā tau matua. Ko te whakakotahitanga o ēnei pūtake rerekē ki tētahi ariā ōrite o ngā rōpū i tīmata ki te Traité des substitutions et des équations algébriques (1870) a Camille Jordan. I whakauruhia e Walther von Dyck (1882) te ariā o te tautuhi i tētahi rōpū mā te whakamahi i ngā kaihanga me ngā pātahitanga, ā, ko ia hoki te tuatahi ki te tuku i tētahi whakamāramatanga arorau o tētahi "rōpū ātaahua", i roto i te kupu o taua wā. Mai i te rau tau 20, ko ngā rōpū...  Pātai: He aha te whāinga o te rangahau?
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