Left coset

Left coset
Tag	`05FD317V2W`
URI	`m2://05FD317V2W`
Kind	definition
Writing	W0 stub
MSC	20A05
Defines	Left coset
Uses	Subgroup

Markdown · JSON · Edit

Definition

Let $G$ be a group, let $H$ be a subgroup of $G$ , and let $g \in G$ . The left coset of $H$ with representative $g$ is the set $gH = \{gh : h \in H\}$ .

Remarks

Two left cosets $aH$ and $bH$ are equal when $a^{-1}b \in H$ , and they are disjoint otherwise. Thus the left cosets of $H$ form a partition of $G$ , and $H = eH$ is one of them. In $(\mathbb{Z}, +)$ , the left cosets of $3\mathbb{Z}$ are $3\mathbb{Z}$ , $1 + 3\mathbb{Z}$ and $2 + 3\mathbb{Z}$ .

Explanations

L1 human

A left coset is a copy of the subgroup that you move by multiplying with a fixed element on the left. Different elements can give the same coset. For the even integers inside the integers, there are two cosets: the even integers and the odd integers.

What this node needs for W1

an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 3 nodes in this order. The foundations come first, and the list ends with this node.

Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.

no evidence level

Group concept
Subgroup definition
Left coset definition

What depends on this

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

A finite group is nilpotent iff it is the product of its Sylow subgroups 00Y9N80VSS
Normality via left and right cosets 01E7SWYJX3
Extensions of solvable groups are solvable 01EY0EM7RP
Quotients of solvable groups are solvable 027ZQS04SJ
Order of a group is the index times the order of the subgroup 04DC0SRN2K
Index of a subgroup 04QEBF8S49
Inner automorphism group is isomorphic to G modulo its centre 05HYGWFG63
Subgroups of solvable groups are solvable 05Y0F9M0TS
Finite p-groups are nilpotent 081WTKYX6C
Class equation 08BPSC6YTE
Product formula for the order of HK 08PY4GNAPK
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
Action on left cosets 0AZG100DKH
First isomorphism theorem 0CXXJNG5B6
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
Universal property of the quotient group 0GJ83E78A5
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
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
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
Order of the alternating group 0SZ1KMNWVE
Third isomorphism theorem 0TJ1V9DDPC
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
Tower law for indices 0ZQJ7KEFQ5

Sources

Web https://en.wikipedia.org/wiki/Coset, Definition
Web https://en.wikipedia.org/wiki/Coset, Properties

What links here

Used by

Normality via left and right cosets 01E7SWYJX3
Index of a subgroup 04QEBF8S49
Action on left cosets 0AZG100DKH
Cosets have the same cardinality as the subgroup 0DKMBWTD7P
Orbit-stabiliser theorem 0E2HSCKZ6C
Quotient group 0YYWCNTPWR