Normal subgroup

Normal subgroup
Tag	`0W77CVBCHQ`
URI	`m2://0W77CVBCHQ`
Kind	definition
Writing	W1 stated
MSC	20A05
Defines	Normal subgroup
Uses	Subgroup, Conjugate subgroup

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Definition

Let $G$ be a group and let $N$ be a subgroup of $G$ . The subgroup $N$ is normal in $G$ when $gng^{-1} \in N$ for all $g \in G$ and all $n \in N$ . The notation is $N \trianglelefteq G$ .

Remarks

Equivalently, $gNg^{-1} = N$ for each $g \in G$ , that is, $N$ equals each of its conjugates. Each group $G$ has the normal subgroups $\{e\}$ and $G$ . In the symmetric group $S_3$ , the subgroup $\{e, (1\,2\,3), (1\,3\,2)\}$ is normal, but the subgroup $\{e, (1\,2)\}$ is not.

Explanations

L1 human

A normal subgroup does not move when you conjugate it by any element of the group. This is the condition that lets you divide the group by the subgroup and get a new group. In an abelian group, every subgroup is normal.

What this node needs for W2

edge instance_of in: one example (0 of 1)
edge not_instance_of in: one non-example that fails one clause (0 of 1)

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What this needs

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Group concept
Conjugate elements definition
Subgroup definition
Conjugate subgroup definition
Normal subgroup definition

What depends on this

67 nodes use this node, directly or through other nodes.

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
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
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
Action on left cosets 0AZG100DKH
Simple group 0C093D1T4M
The kernel is a normal subgroup 0C9ZKYSPYR
First isomorphism theorem 0CXXJNG5B6
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
Abelianisation 0H9K5D3RDV
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
Second isomorphism theorem 0PPP2J2DJV
Internal direct product 0Q2MVQ7JMT
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
Groups of order pq 0YBR6QPY4Y
Integers modulo n as a quotient group 0YJ3WRR9RY
Free abelian group 0YMSEZE1VH
Quotient group 0YYWCNTPWR
Burnside's lemma 0Z47M1B2VS

Sources

Web https://en.wikipedia.org/wiki/Normal_subgroup, Definitions

What links here

Used by

Semidirect product 014E1DZQE8
Normality via left and right cosets 01E7SWYJX3
Subnormal series 096HN851YZ
Simplicity of the alternating group 0ARPRJQWZ2
Simple group 0C093D1T4M
The kernel is a normal subgroup 0C9ZKYSPYR
Nilpotent group 0DZPD2MG33
A subgroup whose index is the smallest prime divisor of the order is normal 0FEBE9NC0E
Subgroups of abelian groups are normal 0JPDTEDCFD
A quotient is abelian iff the subgroup contains the commutator subgroup 0MS8QGA6S5
Internal direct product 0Q2MVQ7JMT
Presentation of a group 0QZCY9PE3Q
Correspondence theorem 0S9ZVZ34MD
Characteristic subgroups are normal 0SRE7K4E49
Normal core 0T0KMZPYV9
Product of a subgroup and a normal subgroup is a subgroup 0TXWV1JGG1
Quotient group 0YYWCNTPWR