Basic Definitions and Results Flashcards
General Linear Group
Centre (of a group)
Quaternion Group
Multiplication determined by:
i2=-1=j2, ij=k=-ji
Cayley’s Theorem Corollary
Simple Facts about Cosets
(4 facts)
Lagrange’s Theorem
Normal Subgroup
Normal subgroups and cosets
Three examples of normal subgroups
Simple group
Products of Subgroups
Normal subgroups
Special linear group of degree n
Universal property for quotients
Homomorphism Theorem
Isomorphism Theorem
Special case of Correspondence Theorem
Direct Product
Two propositions for determining if a group is the direct product of two subgroups
Proposition to determine if a group is the direct product of n subgroups
Example for corollary:
Theorem: Classification of groups
Summary: finite groups
Character of a group
Quadratic residue modulo p
Dual group (and theorem)
Orthogonality Relations (character of a group)
The order of ab
For any map of sets alpha:X –> G from X to a group G, there exists a unique homomorphism FX –> G making the following diagram commute:
Universal property of the inclusion X–>FX characterizes it
Let G be the group defined by the presentation (X,R). For any group H and map of sets alpha:X –> H sending each element of R to 1 (in the obvious sense), there exists a unique homomorphism G –> H making the following diagram commute
