Extension material

Question 1

commutator

Answer 1

[x,y] = x^-1 y^-1 x y

Question 2

[x,y] = 1 <=>

Answer 2

xy=yx

Question 3

If x^g = y^h

Answer 3

then x^(gh^1) = y

Question 4

derived subgroup of G

Answer 4

G’ = [G,G]

Question 5

G’’ =

Answer 5

G’’= [G’,G’]

Question 6

derived series of G

Answer 6

G= G^(0) > G^(1) > G^(3) >…

Question 7

G^(n) =

Answer 7

G^(0) = G
G^(n) = (G^(n-1))' = [G^(n-1), G^(n-1)]

Question 8

G is a soluble group

Answer 8

G is a soluble group iff G^(n) =1 for some n

Question 9

what is the derived length of G

Answer 9

the least such n such that G is soluble

Question 10

lower central series of G

Answer 10

γ1(G0) ; γ2(G) = [G,G]

γi(G) = [γi-1(G),G] for i>2

Question 11

upper central series of G

Answer 11

Z0(G) =1; Z1(G= Z(G)
Zi(G) is the inverse image in G of Z(G/Zi-1(G)) for i>1

Zi-1(G)

Question 12

G is nilpotent <=>

Answer 12

γm(G)=1 for some m. If n+1 is the least such m, then n is called the nilpotency class of G

Question 13

If G is nilpotent.

Answer 13

Then all subgroups of G are nilpotent and if N is a normal subgroup of G, then G/N is nilpotent

Question 14

Every p group

Answer 14

is nilpotent

Question 15

abelian, nilpotent and soluble relationshop

Answer 15

abelian not a subset of nilpotent, not a subset of soluble