- #1

- 146

- 4

## Homework Statement

Suppose [itex]N \lhd G [/itex] and [itex]K \vartriangleleft G[/itex] and [itex]N \cap K = \{e\}[/itex]. Show that if

[itex]n \in N [/itex]and [itex]k \in K[/itex], then [itex]nk = kn[/itex]. Hint: [itex]nk = kn[/itex] if and

only if [itex]nkn^{-1}k^{-1} = e[/itex].

## Homework Equations

These "relevant equations" were not provided with the problem I'm just putting them here to make my solution more clear.

[itex]e=k_1^{-1}k_1[/itex]

[itex]e=n_1^{-1}n_1[/itex]

## The Attempt at a Solution

Let [itex]n_1,n_2\in N[/itex] and let [itex]k_1,k_2\in K[/itex]

Then

[itex](n_1)(k_1)(n_2)(k_2)=(n_1)(k_1)(n_2)e(k_2)=n_1(k_1n_2k_1^{-1})(k1k2)=n1NK=NK[/itex]

But

[itex](n_1)(k_1)(n_2)(k_2)=(n_1)(k_1)e(n_2)(k_2)=(n_1k_1n_1^{-1})(n_1n_2)k_2=KNk_2=Kk_2N=KN[/itex] where we used the fact that in this case, [itex]k_2 \notin N[/itex].

Therefore [itex]NK=KN[/itex] for all [itex]n\in N[/itex] and [itex]k\in K[/itex]

This seems correct to me but I didn't use the hint and my usage of [itex]N \cap K = \{e\}[/itex] seems a little hand wavey.

Please help.