Chapter 2 SUBGROUPS Flashcards
1
Q
A^g = …
A
A^g = {g^-1 ag | a∈A}
2
Q
A^gh = ….
A
A^gh = (A^g)^h
3
Q
what is centralizer of S in G
A
CG(S)={g ∈ G|gx=xg ∀ x in S}
4
Q
what is normalizer of S in G
A
NG(S)={g ∈ G| S^g = S}
5
Q
what is subgroup of G generated by S
A
= {x1,….,xm | m ∈ N, xi ∈ SUS^-}
6
Q
what is a conjugate in G of S
A
S^g={g^-1 x g | x ∈ S}
7
Q
how is the centralizer, normalizer and related to G
A
they are all subgroups of G
8
Q
Centralizer is a subset of the normalizer!
A
yep
9
Q
S is a subset and therefore a subgroup of the normalizer!
A
yep
10
Q
what is the centre of G? Z(G)?
A
Z(G)=CG(G), so Z(G)≤G
11
Q
IF H≤G, K≤G, then what does HK≤G mean?
A
HK≤G iff HK=KH
12
Q
|H K| =?
A
|H K| = |H| |K| / |H∩K|
13
Q
If |G| = |H| |K| / |H∩K|, then G = …?
A
then G = HK
14
Q
Dih(2n) is …?
A
a group of order 2n where * is matrix multiplication
