Basic Definitions and Results Flashcards
(32 cards)
1
Q
General Linear Group
A
2
Q
Centre (of a group)
A
3
Q
Quaternion Group
A
Multiplication determined by:
i2=-1=j2, ij=k=-ji
4
Q
Cayley’s Theorem Corollary
A
5
Q
Simple Facts about Cosets
(4 facts)
A
6
Q
Lagrange’s Theorem
A
7
Q
Normal Subgroup
A
8
Q
Normal subgroups and cosets
A
9
Q
Three examples of normal subgroups
A
10
Q
Simple group
A
11
Q
Products of Subgroups
A
12
Q
Normal subgroups
A
13
Q
Special linear group of degree n
A
14
Q
Universal property for quotients
A
15
Q
Homomorphism Theorem
A
16
Q
Isomorphism Theorem
A
17
Q
A
18
Q
Special case of Correspondence Theorem
A
19
Q
Direct Product
A
20
Q
Two propositions for determining if a group is the direct product of two subgroups
A
21
Q
Proposition to determine if a group is the direct product of n subgroups
A
22
Q
Example for corollary:
A
23
Q
Theorem: Classification of groups
A
24
Q
Summary: finite groups
A
25
Character of a group
26
Quadratic residue modulo p
27
Dual group (and theorem)
28
Orthogonality Relations (character of a group)
29
The order of ab
30
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:
31
Universal property of the inclusion X--\>FX characterizes it
32
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
