Group theory and generalizations Foundations Concept

Left coset

Summary

A left coset of a subgroup $H$ of a group $G$ is a set $gH$ , with $g \in G$ . You get it when you multiply each element of $H$ on the left by $g$ . The left cosets of $H$ partition $G$ .

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36 pages build on this one.

A finite group is nilpotent iff it is the product of its Sylow subgroups
Groups of prime order are cyclic
Order of an element divides the order of the group
Fixed points of a p-group action
Second Sylow theorem
Groups of order p squared are abelian
The number of Sylow subgroups is the index of the normaliser
The alternating group A5 is simple
Abelian simple groups have prime order
Cauchy's theorem
The number of Sylow subgroups divides the index
Converse of Lagrange's theorem fails for the alternating group A4
Third Sylow theorem
Frattini argument
Size of an orbit times the size of the stabiliser
Subgroups of index two are normal
First Sylow theorem
Groups of order pq
Burnside's lemma
g to the order of the group is the identity
Lagrange's theorem
Galois group of a prime degree polynomial with two nonreal roots
Proof that subgroups of index two are normal
Induced representation of a locally compact group
Nichols–Zoeller theorem
Hall's theorem on solvable groups
Induced representation
Quotient of a topological group
Proof that the order of an element divides the order of the group
Open subgroups are closed
Hall subgroup
Sylow theorems
Frattini subgroup
Proof that groups of prime order are cyclic
Transfer homomorphism
Proof of Lagrange's theorem

Tag	`03DEH43F4H`
URI	`m2://03DEH43F4H`
Kind	concept
Writing	W0 stub
MSC	20A05

Left coset

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