- #1

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## Homework Statement

Let ##H, K, N## be nontrivial normal subgroups of a group ##G## and suppose that ##G = H \times K##. Prove that ##N## is in the center of ##G## or ##N## intersects one of ##H,K## nontrivially

## Homework Equations

## The Attempt at a Solution

I presume that ##G = H \times K## means that ##G## is the internal direct product of the subgroups of ##H## and ##K##. So ##G## being the internal direct product of ##G## means that ##G= \langle H \cup K \rangle## with ##H \cap K = \{e\}##; and since at least one of the two subgroups is normal, ##\langle H \cup K \rangle = HK## (Is this also called the internal weak direct product?) Now, suppose that ##N## intersects ##H## and ##K## trivially. Then ##nh = hn## for all ##n \in N## and ##h \in H## ; and a similar commutation relation holds for ##K##. Then given ##g \in HK##, which means ##g = hk## for some ##h \in H## and ##k \in K##, we have that ##nhk = hkn##, thereby showing ##n \in Z(G)##.

Does this sound right?