# Cyclic Group Homomorphisms and Subgroups

1. Let G be a finite group with a normal subgroup N. Prove that if K is a subgroup of $$G$$ such that $$K ∩ N = {e}$$ then $$|K||[G : N]$$
2. Let G be a cyclic group and $$f : G → G$$ a function. Prove that $$f$$ is a homomorphism iff there is an integer $$k$$ such that $$f(a) = a^{k}$$ for every $$a ∈ G$$.

For the first question would the second iso thm be used here? Since $$KN/N$$ isomorphic to $$K/K ∩ N$$ can be reduced to by lagrange's thm |KN/N|=|K| but I am not sure what I would do after this.

For the second question this looks familar to the defintion of a subgroup of a cyclic group but I am not sure how to approach it..

Since $$N$$ is normal in $$G$$ the set $$KN$$ is a subgroup of $$G$$. Now, $$|KN|=\frac{|K||N|}{|K\cap N|}=|K||N|$$. Now the index of the subgroup $$KN$$ in $$G$$ is $$[G:KN]=\frac{|G|}{|KN|}=\frac{1}{|K|}\frac{|G|}{|N|}\implies |K|[G:KN]=[G:N]\implies|K|\bigg| [G:N].$$

Next if $$f$$ is a group homomophism then $$f(a)=a^k$$ for each $$a\in G$$ as $$G=\langle g\rangle\implies f(g)=g^n$$ for some $$n\in \Bbb Z$$. So that, $$f(g^k)=g^{nk}$$ for all $$k\in\Bbb Z$$.

Now suppose we have a fixed integer $$k$$ for which, $$f(a)=a^{k}$$ for each $$a\in G$$. Then clearly $$f(a\cdot b)=(ab)^k=a^kb^k=f(a)f(b),\forall a,b\in G$$ as cyclic groups are abelian. So $$f$$ is a group homomorphism.

By the third isomorphism theorem, you have that every subgroup of $$G/N$$ is of the form $$H/N$$, for some subgroup $$H$$ in $$G$$.

So all you need to prove is that $$KN$$ is a group itself (from which it follows it is a subgroup of $$G$$, not just a subset). You might know already that $$KN$$ is a group of $$G$$ iff $$KN=NK$$, and this follows since $$N$$ is a normal subgroup.

You therefore have the required result by Lagrange's theorem.

Hint: For the second question, write $$f(g)=g^k$$, where $$G=\langle g \rangle$$.