Subgroup

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• A finite group is nilpotent iff it is the product of its Sylow subgroups 00Y9N80VSS
• Image of a homomorphism 0112DTS464
• Semidirect product 014E1DZQE8
• Characteristic subgroup 01CS8Q2APC
• Normality via left and right cosets 01E7SWYJX3
• Extensions of solvable groups are solvable 01EY0EM7RP
• Quotients of solvable groups are solvable 027ZQS04SJ
• Infinite cyclic group 02B1T50JD1
• Order of a cyclic subgroup equals the order of its generator 031M8HX7DA
• Subgroups of a finite cyclic group correspond to divisors of its order 0423TXR9AN
• Order of a group is the index times the order of the subgroup 04DC0SRN2K
• Index of a subgroup 04QEBF8S49
• Cyclic groups are abelian 04RG02S6BA
• The dihedral group of order 8 054SBCQ93T
• Stabilisers of points in the same orbit are conjugate 05BM304NRX
• Left coset 05FD317V2W
• Inner automorphism group is isomorphic to G modulo its centre 05HYGWFG63
• Groups of prime order are cyclic 05MM4EK4SA
• Subgroup criterion 05N4TQKPH0
• Subgroups of solvable groups are solvable 05Y0F9M0TS
• Right coset 060Z37VVB6
• Classification of finite simple groups 06J6P03MG7
• Cyclic groups of the same order are isomorphic 0736AXH2ME
• Classification of finite abelian groups 07CQR9BW3R
• Finite p-groups are nilpotent 081WTKYX6C
• Class equation 08BPSC6YTE
• Product of cyclic groups is cyclic iff the orders are coprime 08FDFEF8D1
• Fixed points of an action 08JF1K4DW1
• Lower central series 08PA437800
• Product formula for the order of HK 08PY4GNAPK
• Commutator subgroup 08SJ5K2CBG
• Order of an element divides the order of the group 08V7JHS3QZ
• Subnormal series 096HN851YZ
• Fixed points of a p-group action 098D2E7DEH
• Second Sylow theorem 09GCQ5STVJ
• Dihedral group 09GK3DE0WV
• 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
• The symmetric group is generated by transpositions 0AQNX7G443
• Simplicity of the alternating group 0ARPRJQWZ2
• Normaliser 0AT1ZSNM25
• Action on left cosets 0AZG100DKH
• Cyclic group 0B85Q41RP6
• Lattice of subgroups 0B8BW4YWTF
• Order of the dihedral group 0BZVNWJ352
• Simple group 0C093D1T4M
• The kernel is a normal subgroup 0C9ZKYSPYR
• First isomorphism theorem 0CXXJNG5B6
• Conjugate subgroup 0D6WHTTH7T
• Upper central series 0DJ9Y9V610
• Cosets have the same cardinality as the subgroup 0DKMBWTD7P
• 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
• Cyclic group of order n 0G0B3DRG65
• Automorphism group of a cyclic group 0G4C8SJ6PV
• Universal property of the quotient group 0GJ83E78A5
• The alternating group A5 is simple 0GKQKF0FXJ
• Nilpotent groups are solvable 0GNG2KVPZJ
• Abelianisation 0H9K5D3RDV
• Finitely generated group 0HY15D6WTX
• 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
• Sylow p-subgroup 0JWZX6FKSH
• Subgroups of cyclic groups are cyclic 0KB1ND7XDS
• The alternating group is generated by three-cycles 0KFZ0QX83G
• The number of Sylow subgroups divides the index 0M3PYJ3TSE
• Generators of a group 0MBV2VHMXA
• A finite abelian group is cyclic iff its exponent equals its order 0MC0AQZSHE
• A quotient is abelian iff the subgroup contains the commutator subgroup 0MS8QGA6S5
• Composition series 0NB8VPB93V
• Product of two subgroups 0NS1YTGM19
• Solvable group 0P3KZK0FTV
• Action by conjugation 0P8CY9V437
• Second isomorphism theorem 0PPP2J2DJV
• Centraliser 0Q0Z1BEXTF
• Internal direct product 0Q2MVQ7JMT
• Third Sylow theorem 0Q8DGAZSJ4
• Von Dyck's theorem 0QPYHHNRXC
• The group of units of Z/pZ is cyclic 0QTHB6PE52
• Frattini argument 0QTZDKJEDV
• Presentation of a group 0QZCY9PE3Q
• Abelian groups are solvable 0R0X73XHAT
• Derived series 0R3H35G96G
• Composition factors 0RFP7F3SEW
• Subgroup generated by a subset 0RG9SPQSWM
• Stabiliser 0RGGF8ZS6D
• 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
• Subgroups of the integers 0SRATCFF1W
• 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
• Number of elements of each order in a cyclic group 0XCYGCCX8V
• Groups of order pq 0YBR6QPY4Y
• Cayley graph 0YE597NHGB
• Integers modulo n as a quotient group 0YJ3WRR9RY
• Free abelian group 0YMSEZE1VH
• Cyclic subgroup generated by an element 0YT586TKXF
• Quotient group 0YYWCNTPWR
• Burnside's lemma 0Z47M1B2VS
• g to the order of the group is the identity 0ZEPNS9BXK
• Tower law for indices 0ZQJ7KEFQ5
• Intersection of subgroups is a subgroup 0ZTGMDNVTB