Conjugate elements

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• A finite group is nilpotent iff it is the product of its Sylow subgroups 00Y9N80VSS
• Semidirect product 014E1DZQE8
• Normality via left and right cosets 01E7SWYJX3
• Extensions of solvable groups are solvable 01EY0EM7RP
• Quotients of solvable groups are solvable 027ZQS04SJ
• Stabilisers of points in the same orbit are conjugate 05BM304NRX
• Inner automorphism group is isomorphic to G modulo its centre 05HYGWFG63
• Subgroups of solvable groups are solvable 05Y0F9M0TS
• Classification of finite simple groups 06J6P03MG7
• Finite p-groups are nilpotent 081WTKYX6C
• Class equation 08BPSC6YTE
• Subnormal series 096HN851YZ
• Fixed points of a p-group action 098D2E7DEH
• Second Sylow theorem 09GCQ5STVJ
• The symmetric group S3 09H9Y0Q11W
• Outer automorphism group 09HEXDQ0CQ
• Groups of order p squared are abelian 09JZ9P4899
• The number of Sylow subgroups is the index of the normaliser 09ZJ6CAK7S
• If G modulo its centre is cyclic then G is abelian 0AH9Q6YNMT
• Simplicity of the alternating group 0ARPRJQWZ2
• Normaliser 0AT1ZSNM25
• Action on left cosets 0AZG100DKH
• Simple group 0C093D1T4M
• The kernel is a normal subgroup 0C9ZKYSPYR
• First isomorphism theorem 0CXXJNG5B6
• Conjugate subgroup 0D6WHTTH7T
• Upper central series 0DJ9Y9V610
• Nilpotent group 0DZPD2MG33
• Jordan–Hölder theorem 0E088ZP533
• Orbit-stabiliser theorem 0E2HSCKZ6C
• Universal property of the abelianisation 0EJ4PC8NJ6
• A subgroup whose index is the smallest prime divisor of the order is normal 0FEBE9NC0E
• Burnside's p^a q^b theorem 0FMAQSB1SC
• Universal property of the quotient group 0GJ83E78A5
• The alternating group A5 is simple 0GKQKF0FXJ
• Nilpotent groups are solvable 0GNG2KVPZJ
• Inner automorphism 0H47DMFN3S
• Abelianisation 0H9K5D3RDV
• Conjugacy classes of the symmetric group correspond to cycle types 0J438GAC8K
• The symmetric group on five or more letters is not solvable 0J53NE674D
• Abelian simple groups have prime order 0JJV80RXHD
• Cauchy's theorem 0JM7PA288B
• Subgroups of abelian groups are normal 0JPDTEDCFD
• The number of Sylow subgroups divides the index 0M3PYJ3TSE
• A quotient is abelian iff the subgroup contains the commutator subgroup 0MS8QGA6S5
• Composition series 0NB8VPB93V
• Solvable group 0P3KZK0FTV
• Action by conjugation 0P8CY9V437
• Second isomorphism theorem 0PPP2J2DJV
• Internal direct product 0Q2MVQ7JMT
• Conjugacy class 0Q88KQD17A
• Third Sylow theorem 0Q8DGAZSJ4
• Von Dyck's theorem 0QPYHHNRXC
• Frattini argument 0QTZDKJEDV
• Presentation of a group 0QZCY9PE3Q
• Abelian groups are solvable 0R0X73XHAT
• Composition factors 0RFP7F3SEW
• Torsion subgroup 0RH1TPDGCQ
• Structure theorem for finitely generated abelian groups 0RY0B72XAS
• A nontrivial finite p-group has nontrivial centre 0RZCTT30H5
• Size of an orbit times the size of the stabiliser 0S19CWFZM2
• Correspondence theorem 0S9ZVZ34MD
• Characteristic subgroups are normal 0SRE7K4E49
• Order of the alternating group 0SZ1KMNWVE
• Normal core 0T0KMZPYV9
• Third isomorphism theorem 0TJ1V9DDPC
• Product of a subgroup and a normal subgroup is a subgroup 0TXWV1JGG1
• Subgroups of index two are normal 0V547YQRMV
• First Sylow theorem 0V98WY94B8
• Size of a conjugacy class is the index of the centraliser 0VK5V9P8G3
• Normal subgroup 0W77CVBCHQ
• Groups of order pq 0YBR6QPY4Y
• Integers modulo n as a quotient group 0YJ3WRR9RY
• Free abelian group 0YMSEZE1VH
• Quotient group 0YYWCNTPWR
• Burnside's lemma 0Z47M1B2VS