Question

Suppose that f is a homomorphism from s4 onto z2. Determine ker f. Determine all homomorphisms from s4 to z2.

Answer 1

φ

(

xy

) = (

xyH

,

xyK

) = (

xH

,

xK

)(

yH

,

yK

) =

φ

(

x

)

φ

(

y

)

.

Thus,

φ

is a homomorphism with

Ker

(

φ

) =

{

g

∈

G

|

(

gH

,

gK

) = (

H

,

K

)

}

=

H

∩

K

=

{

e

}

.

By the First Isomorphism Theorem,

G

=

G

/

{

e

}

=

G

/

Ker

(

φ

)

≈

φ

(

G

/

H

⊕

G

/

K

)

⊆

G

/

H

⊕

G

/

K

.