Are π1 and π2 True Group Homomorphisms with Identifiable Kernels and Images?

[kathrynag]

Homework Statement

For groups G1 and G2, let p1 : G1 × G2 → G1 be defined by p1((g1, g2)) = g1 and let
p2 : G1 × G2 → G2 be defined by p2((g1, g2)) = g2. Show that p1 and p2 are group
homomorphisms and determine the kernel and image of each.

Homework Equations

The Attempt at a Solution

I need to show p1(a,b)p1(c,d)=p1(ab,cd)
=ac?

 

[Outlined]

yes that is correct, note you almost have the answer.

When you construct a new group G₁ × G₂ the group operations becomes (well most of the time) (g₁, g₂)(r₁, r₂) = (g₁r₁, g₂r₂)

kind request to use SUP, SUB and TEX tags to make messages easier to read to the people here.

 

[kathrynag]

kernel we want p1(a)=e
=(e,g1),(e,g2)

 

[Outlined]

the kernel of p₁ is {e} x G₂

again the request to use the right tags.

 

[kathrynag]

kernel of p2 is {e} x G_{1}

Now I have no clue on image

 

[Outlined]

no, it is not

 

[kathrynag]

Is it G_{2} x {e}

 

[Outlined]

Your map was from G₁ x G₂ to G₁

 

[kathrynag]

Then it's G_{1} x {e}