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## Main Question or Discussion Point

If H and N are subgroups of a group G. And we define [tex]HN = \{ hy | h \in H, y \in N \}[/tex],

Then I know that the following are true:

But does anybody know the proof to any of them?

For 2 I know that if they are both normal, then [tex]G=HN[/tex] and [tex]H \cap N = \{ e \}[/tex], but really I don't know how to prove it.

Any help or suggestions is appreciated.

Then I know that the following are true:

- If [tex]N[/tex] is a normal subgroup. Then [tex]HN[/tex] is a subgroup of [tex]G[/tex].

- If [tex]H[/tex] and [tex]N[/tex] are both normal subgroups. Then [tex]HN[/tex] is normal.

But does anybody know the proof to any of them?

For 2 I know that if they are both normal, then [tex]G=HN[/tex] and [tex]H \cap N = \{ e \}[/tex], but really I don't know how to prove it.

Any help or suggestions is appreciated.