20 Group theory and generalizations

See MSC 20 on zbMATH Open.

Name	Tag	Kind	MSC
Sylow p-subgroup	0053CQE9Y2	concept	20D20
Laws of exponents in a group	00DRK90Z6K	statement	20A05
Orbit	00F2JFA41J	definition	20B05
Direct product of groups	00HF4RJ5QN	concept	20E34
A finite group is nilpotent iff it is the product of its Sylow subgroups	00Y9N80VSS	statement	20F18, 20D15
Image of a homomorphism	0112DTS464	definition	20A05
Semidirect product	014E1DZQE8	definition	20E22
Cayley's theorem	0151VYVX27	statement	20B30
Characteristic subgroup	01CS8Q2APC	definition	20F28
Normality via left and right cosets	01E7SWYJX3	statement	20A05
Extensions of solvable groups are solvable	01EY0EM7RP	statement	20F16
Free group	020Z5SEN27	definition	20E05
Quotients of solvable groups are solvable	027ZQS04SJ	statement	20F16
Infinite cyclic group	02B1T50JD1	example	20A05
Sign of a permutation	02DKV69ZH0	concept	20B30
Transposition	02DTRYKGE8	definition	20B30
Transitive action	02GK6HZYBT	concept	20B05
Centraliser	02YYV36BFM	concept	20A05
Order of a cyclic subgroup equals the order of its generator	031M8HX7DA	statement	20A05
Left coset	03DEH43F4H	concept	20A05
Subgroups of a finite cyclic group correspond to divisors of its order	0423TXR9AN	statement	20K01
Order of a group is the index times the order of the subgroup	04DC0SRN2K	statement	20D60
Index of a subgroup	04QEBF8S49	definition	20A05
Cyclic groups are abelian	04RG02S6BA	statement	20A05
Symmetric group	0515A6D2Y1	concept	20B30
The dihedral group of order 8	054SBCQ93T	example	20B25, 20D15
Stabilisers of points in the same orbit are conjugate	05BM304NRX	statement	20B05
Left coset	05FD317V2W	definition	20A05
Inner automorphism group is isomorphic to G modulo its centre	05HYGWFG63	statement	20F28
Groups of prime order are cyclic	05MM4EK4SA	statement	20D60
Subgroup criterion	05N4TQKPH0	statement	20A05
Subgroups of solvable groups are solvable	05Y0F9M0TS	statement	20F16
Right coset	060Z37VVB6	definition	20A05
P-group	06DAH3VS6M	definition	20D15
Classification of finite simple groups	06J6P03MG7	statement	20D05
P-group	06QYRTX9KQ	concept	20D15
Commutator of two elements	06RGT4V6Y6	definition	20F12
Abelian group	071DDFVG4J	definition	20Kxx
Cyclic groups of the same order are isomorphic	0736AXH2ME	statement	20A05
Even and odd permutations	0771AJP4BX	definition	20B30
Classification of finite abelian groups	07CQR9BW3R	statement	20K01
Disjoint cycles commute	07E6D8R4D6	statement	20B30
Finite p-groups are nilpotent	081WTKYX6C	statement	20D15, 20F18
Finitely generated group	082DWWXMRJ	concept	20F05
Abelian group	082GHNPJ53	concept	20Kxx
Group isomorphism	085PND4KNT	definition	20A05
Cyclic subgroup generated by an element	08ABT85339	concept	20A05
Class equation	08BPSC6YTE	statement	20E45
Product of cyclic groups is cyclic iff the orders are coprime	08FDFEF8D1	statement	20K01
Group of units of a monoid	08FHNR8Z5D	definition	20M10
Homomorphisms preserve identity and inverses	08FKQD7NJ3	statement	20A05
Fixed points of an action	08JF1K4DW1	definition	20B05
Group isomorphism	08MG7H8JW5	concept	20A05
Lower central series	08PA437800	definition	20F14
Product formula for the order of HK	08PY4GNAPK	statement	20D60
Commutator subgroup	08SJ5K2CBG	definition	20F12
Order of an element divides the order of the group	08V7JHS3QZ	statement	20D60
Subnormal series	096HN851YZ	definition	20E15
Fixed points of a p-group action	098D2E7DEH	statement	20D15, 20B05
Second Sylow theorem	09GCQ5STVJ	statement	20D20
Dihedral group	09GK3DE0WV	definition	20B25
The symmetric group S3	09H9Y0Q11W	example	20B30
Outer automorphism group	09HEXDQ0CQ	definition	20F28
Groups of order p squared are abelian	09JZ9P4899	statement	20D15
The number of Sylow subgroups is the index of the normaliser	09ZJ6CAK7S	statement	20D20
If G modulo its centre is cyclic then G is abelian	0AH9Q6YNMT	statement	20A05
The symmetric group is generated by transpositions	0AQNX7G443	statement	20B30
Abelianisation	0AQZV0905N	concept	20F12
Simplicity of the alternating group	0ARPRJQWZ2	statement	20B30, 20D06
Commutator subgroup	0ASGSTXSBW	concept	20F12
Normaliser	0AT1ZSNM25	definition	20A05
Stabiliser	0AYMP41THN	concept	20B05
Action on left cosets	0AZG100DKH	example	20B05
Faithful action	0B0WE1GKTK	definition	20B05
Cyclic group	0B85Q41RP6	definition	20A05
Order of a group	0B85S2V2DR	concept	20A05
Lattice of subgroups	0B8BW4YWTF	concept	20E15
Cycle type	0BCHJ30REG	definition	20B30
Trivial group	0BNPC9850Y	example	20A05
Additive group of integers	0BSCJA5G53	example	20K99
Order of the dihedral group	0BZVNWJ352	statement	20B25
Simple group	0C093D1T4M	definition	20E32
The kernel is a normal subgroup	0C9ZKYSPYR	statement	20A05
Normal subgroup	0CB0DX5BVV	concept	20A05
Universal property of the free group	0CDWKDYWN6	statement	20E05
Special orthogonal group	0CGYNDCWRR	object	15B10, 20G20
Automorphism group	0CWXCWP0Y0	definition	20F28
First isomorphism theorem	0CXXJNG5B6	statement	20A05
Conjugate elements	0D1VWQGZZ7	definition	20E45
Conjugate subgroup	0D6WHTTH7T	definition	20E45
Uniqueness of inverses	0D9NDGE9KP	statement	20A05
Torsion subgroup	0DBWXQRXQJ	concept	20K10
Conjugacy class	0DEBXAJ5SA	concept	20E45
Order of a permutation is the lcm of its cycle lengths	0DH7HMXP9F	statement	20B30
Upper central series	0DJ9Y9V610	definition	20F14
Cosets have the same cardinality as the subgroup	0DKMBWTD7P	statement	20A05
Index of a subgroup	0DTXBQPA2S	concept	20A05
Conjugate elements	0DXD4QSZ8A	concept	20E45
Centre of a group	0DZ93493VN	concept	20A05
Nilpotent group	0DZPD2MG33	definition	20F18
Jordan–Hölder theorem	0E088ZP533	statement	20E15, 20D30
Orbit-stabiliser theorem	0E2HSCKZ6C	statement	20B05
Cayley table	0E4WH0R7HG	definition	20A05
Universal property of the abelianisation	0EJ4PC8NJ6	statement	20F12
g to the n equals 1 iff the order divides n	0F5QC101Q7	statement	20A05
Group of units of Z/nZ	0F8X8ZRQ53	object	11A07, 20K01
Exponent of a group	0FCCTFA275	definition	20F50
General linear group	0FCHE902D7	object	20G15, 15A09
A subgroup whose index is the smallest prime divisor of the order is normal	0FEBE9NC0E	statement	20D60
Group of units of a monoid	0FJ90Y9926	concept	20M10
Burnside's p^a q^b theorem	0FMAQSB1SC	statement	20D10
Reduced word	0FPXK5CHQ3	definition	20E05
Transposition	0FWVRHBRWM	concept	20B30
Solvable group	0FXJ7YFMMX	concept	20F16
Cyclic group of order n	0G0B3DRG65	example	20K01
Automorphism group of a cyclic group	0G4C8SJ6PV	statement	20F28
Normal core	0GETX79CJN	concept	20D25
Universal property of the quotient group	0GJ83E78A5	statement	20A05
The alternating group A5 is simple	0GKQKF0FXJ	example	20B30, 20D06
Nilpotent groups are solvable	0GNG2KVPZJ	statement	20F18, 20F16
Units of Z/nZ are the residues coprime to n	0GXNHEHQNA	statement	11A07, 20K01
Cayley graph	0GYQK9NDJF	concept	05C25, 20F65
Conjugate subgroup	0H11ZPX4ND	concept	20E45
Semidirect product	0H3WZXCZNF	concept	20E22
Euclidean group	0H3YY8CPKH	object	20H15
Inner automorphism	0H47DMFN3S	definition	20F28
Abelianisation	0H9K5D3RDV	definition	20F12
Group homomorphism	0HV9Y7WG5R	definition	20A05
Finitely generated group	0HY15D6WTX	definition	20F05
Cycle type	0J3K4Y1KPD	concept	20B30
Conjugacy classes of the symmetric group correspond to cycle types	0J438GAC8K	statement	20B30
The symmetric group on five or more letters is not solvable	0J53NE674D	statement	20F16, 20B30
Sign is a homomorphism	0JEJJCB0EX	statement	20B30
Presentation of a group	0JGEQXQ4AC	concept	20F05
Abelian simple groups have prime order	0JJV80RXHD	statement	20E32, 20K01
Cauchy's theorem	0JM7PA288B	statement	20D20
Subgroups of abelian groups are normal	0JPDTEDCFD	statement	20A05
Order of an element	0JPYSD5E0B	definition	20A05
Order of the general linear group over a finite field	0JQ1KNQW2K	statement	20G40
Sylow p-subgroup	0JWZX6FKSH	definition	20D20
Group homomorphism	0JYTG5K9Y4	concept	20A05
Dihedral group	0JZ27DMJ5D	concept	20B25
Automorphism group	0K1SCPBNKJ	concept	20F28
Action by left multiplication	0K5TWJAGPD	example	20B05
Lower central series	0K98R7A4VC	concept	20F14
Characteristic subgroup	0KABPMRHM2	concept	20F28
Subgroups of cyclic groups are cyclic	0KB1ND7XDS	statement	20K27
Reduced word	0KBTP8Q2FS	concept	20E05
The alternating group is generated by three-cycles	0KFZ0QX83G	statement	20B30
Free group	0KTW26C80J	concept	20E05
Cyclic group	0KVGH0J5B3	concept	20A05
Quotient group	0KYX2N69J0	concept	20A05
The number of Sylow subgroups divides the index	0M3PYJ3TSE	statement	20D20
Generators of a group	0MBV2VHMXA	concept	20F05
A finite abelian group is cyclic iff its exponent equals its order	0MC0AQZSHE	statement	20K01
A quotient is abelian iff the subgroup contains the commutator subgroup	0MS8QGA6S5	statement	20F12
Subgroup	0N0PXCTYA7	definition	20A05
A homomorphism is injective iff its kernel is trivial	0N2P0XRTDF	statement	20A05
Monoid	0N6R902FG1	concept	20M10
Composition series	0NB8VPB93V	definition	20E15, 20D30
Right coset	0NE1DK7H4T	concept	20A05
Quaternion group	0NRJGW864D	object	20D15
Product of two subgroups	0NS1YTGM19	definition	20A05
Solvable group	0P3KZK0FTV	definition	20F16
Faithful action	0P7G9NGG2G	concept	20B05
Action by conjugation	0P8CY9V437	example	20E45, 20B05
Order of the symmetric group	0PCKBBMEXC	statement	20B30
Converse of Lagrange's theorem fails for the alternating group A4	0PDBVPP233	counterexample	20D60
Composition factors	0PHTC93M58	concept	20E15, 20D30
Klein four-group	0PMRQJVTME	object	20K01
Order of an inverse	0PMT9YJZVW	statement	20A05
Second isomorphism theorem	0PPP2J2DJV	statement	20A05
Product of two subgroups	0PT52XQZD7	concept	20A05
Commutator of two elements	0PXW0BN7DW	concept	20F12
Centraliser	0Q0Z1BEXTF	definition	20A05
Internal direct product	0Q2MVQ7JMT	definition	20E34
Derived series	0Q5G5DJ4P2	concept	20F14
Nilpotent group	0Q7MMQHC1P	concept	20F18
Conjugacy class	0Q88KQD17A	definition	20E45
Third Sylow theorem	0Q8DGAZSJ4	statement	20D20
Outer automorphism group	0Q8Y862DTZ	concept	20F28
Transitive action	0QP2YYR79N	definition	20B05
Von Dyck's theorem	0QPYHHNRXC	statement	20F05
The group of units of Z/pZ is cyclic	0QTHB6PE52	statement	11A07, 20K01
Frattini argument	0QTZDKJEDV	statement	20D20
Presentation of a group	0QZCY9PE3Q	definition	20F05
Abelian groups are solvable	0R0X73XHAT	statement	20F16
Derived series	0R3H35G96G	definition	20F14
Monoid	0R47MAD57S	definition	20M10
Unitary group	0R9R0Y1HDK	object	15B57, 20G20
Composition factors	0RFP7F3SEW	definition	20E15, 20D30
Subgroup generated by a subset	0RG9SPQSWM	definition	20A05
Stabiliser	0RGGF8ZS6D	definition	20B05
Torsion subgroup	0RH1TPDGCQ	definition	20K10
Symmetric group	0RJMNEN3BJ	definition	20B30
Inner automorphism	0RN00NTM3S	concept	20F28
Subgroup generated by a subset	0RP7VWQ0YN	concept	20A05
Fixed points of an action	0RRWNMYRXC	concept	20B05
Structure theorem for finitely generated abelian groups	0RY0B72XAS	statement	20K25
A nontrivial finite p-group has nontrivial centre	0RZCTT30H5	statement	20D15
Size of an orbit times the size of the stabiliser	0S19CWFZM2	statement	20B05
Image of a homomorphism	0S21GNW9ZB	concept	20A05
Centre of a group	0S7Y9Q990Q	definition	20A05
Correspondence theorem	0S9ZVZ34MD	statement	20A05
Order of a group	0SEX9F9J5Z	definition	20A05
Sign of a permutation	0SK90238BH	definition	20B30
Even and odd permutations	0SQTMQTSRZ	concept	20B30
Subgroups of the integers	0SRATCFF1W	statement	20K27
Characteristic subgroups are normal	0SRE7K4E49	statement	20F28
Order of the alternating group	0SZ1KMNWVE	statement	20B30
Normal core	0T0KMZPYV9	definition	20D25
Alternating group	0T13HM377F	definition	20B30
Third isomorphism theorem	0TJ1V9DDPC	statement	20A05
Orthogonal group	0TKYAEJ8N7	object	15B10, 20G20
Product of a subgroup and a normal subgroup is a subgroup	0TXWV1JGG1	statement	20A05
Sign of a cycle	0V0E04DWX2	statement	20B30
Semigroup	0V3CFG1KK9	definition	20M10
Subgroups of index two are normal	0V547YQRMV	statement	20A05
First Sylow theorem	0V98WY94B8	statement	20D20
Special linear group	0V9PMN0RBH	object	15A15, 20G15
Internal direct product	0VBZH90R1A	concept	20E34
Size of a conjugacy class is the index of the centraliser	0VK5V9P8G3	statement	20E45
Cancellation laws	0VSNX4P2Q0	statement	20A05
Order of a power	0VVPQNVM8S	statement	20A05
Normal subgroup	0W77CVBCHQ	definition	20A05
Kernel of a homomorphism	0W88CEGQP1	concept	20A05
Group action	0WCY42EV9V	concept	20B05
Order of an element	0WE2NHZJ7B	concept	20A05
Invariant factors and elementary divisors	0WHCA4SVP2	concept	20K01, 20K25
Free abelian group	0WJX8MGG37	concept	20K15, 20K20
Group of nth roots of unity	0WX11EA3F3	object	20K01
Subgroup	0X1JXT47GA	concept	20A05
Number of elements of each order in a cyclic group	0XCYGCCX8V	statement	20K01
Decomposition of a permutation into transpositions	0XK6FECMW5	statement	20B30
Semigroup	0XMEM406JM	concept	20M10
Composition series	0XV2A49X6Q	concept	20E15, 20D30
Upper central series	0XW54DH28V	concept	20F14
Cayley table	0YB2NRP0T5	concept	20A05
Groups of order pq	0YBR6QPY4Y	example	20D20
Cayley graph	0YE597NHGB	definition	05C25, 20F65
Normaliser	0YGN3VAA0D	concept	20A05
Integers modulo n as a quotient group	0YJ3WRR9RY	example	20K01
Free abelian group	0YMSEZE1VH	definition	20K15, 20K20
Simple group	0YMYMNQ0QZ	concept	20E32
Direct product of groups	0YP7WYXF9V	definition	20E34
Orbit	0YQPZB5Q1G	concept	20B05
Group action	0YSE802YCQ	definition	20B05
Cyclic subgroup generated by an element	0YT586TKXF	definition	20A05
Decomposition of a permutation into disjoint cycles	0YWDAZPRPQ	statement	20B30
Quotient group	0YYWCNTPWR	definition	20A05
Kernel of a homomorphism	0Z424HK8CQ	definition	20A05
Burnside's lemma	0Z47M1B2VS	statement	20B05, 05E18
Alternating group	0Z5905Q73D	concept	20B30
Orbits partition the set	0ZC7FYE0NZ	statement	20B05
Exponent of a group	0ZDW0H0VX4	concept	20F50
g to the order of the group is the identity	0ZEPNS9BXK	statement	20D60
Symmetry group	0ZG3Z5JZ85	concept	20B25
Cycle notation	0ZKPH2BD1T	concept	20B30
Tower law for indices	0ZQJ7KEFQ5	statement	20A05
Intersection of subgroups is a subgroup	0ZTGMDNVTB	statement	20A05
Subnormal series	0ZZR8D7Y1Z	concept	20E15
Lagrange's theorem	3C53TD	statement	20D60
Group	G5S1M9	concept	20A05