Normal subgroups and homomorphisms

Question 1

normal subgroup if

Answer 1

N^g = N

Question 2

If G is abelian, all subgroups of G are

Answer 2

If G is abelian, all subgroups of G are normal subgroups

Question 3

If G is a group, N≤ G, N≤Z(G) then N is

Answer 3

If G is a group, N≤ G, N≤Z(G) then N is a normal subgroup of G

Question 4

If G a group N≤G, [G:N] = 2, then…

Answer 4

If G a group N≤G, [G:N] = 2, then N is a normal subgroup of G

Question 5

G/N =

Answer 5

G/N = {Ng | g in G} = {gbar| g in G}

Question 6

If G/ Z(G) is cyclic, then

Answer 6

If G/ Z(G) is cyclic, then G is abelian

Question 7

H is a normal subgroup of G <=>

Answer 7

H/N is a normal subgroup of G/N

Question 8

homomorphism

Answer 8

(g1g2)θ = (g1θ)(g2θ)

Question 9

|G/N| =

Answer 9

[G:N] = |G|/|N|

Question 10

First isomorphism theorem

Answer 10

G/ker(theta) is isomorphic to Gtheta