I am working on the semidirect product in group theory. And I got stuck on a question which is not difficult. Here is the question:

IfandQ, K are groupsis ah: Q -->Aut(K)homomorphism

Prove that the semidirect product is a group( sorry, I can't type the notation of semidirect product). So, I have to letG = (K x Q, *, (1,1))

The group operation * is

(k_1,q_1)*(k_2,q_2) =(k_1h(q_1)(k_2), q_1q_2)

This is my attempt.

First show that the group is associative:

[(k_1,q_1)*(k_2,q_2)]*(k_3,q_3) = (k_1h(q_1)(k_2) , q_1q_2)*(k_3,q_3)

= (k_1h(q_1)(k_2)h(q_1q_2)(k_3),q_1q_2q_3)

= ...???

then I dont know how to simplify this. Can we use the fact that h is homomorphism.

And:

(k_1,q_1)*[(k_2,q_2)*(k_3,q_3)] = (k_1,q_1)*(k_2h(q_2)(k_3),q_2q_3)

=(k_1h(q_1)(k_2)h(q_2)(k_3),q_1q_2q_3)

= ......???

Second, show that the group G has an identity,

If: (1,1)*(k,q) = (1.h(1)k,1.q) = (k,q)

and (k,q)*(1,1) = (k.h(q)(1),q.1) = (k.h(q),q) = (???,q)

Finally, G has an inverse if (k,q)*(k^-1,q^-1) = (1,1)

Thus, (k,q)*(k^-1,q^-1) = (k.h(q)(k^-1),q.q^-1) =(h(q), 1)

Then G is a group

I dont know what h(q) is equal to, so that I can simplify.

If you find any mistake please correct it.

Thank you so much

P.S: Sorry about the text, it's a bit hard to read if you write down on paper it will be easier.