I was reading on wikipedia on direct product of groups because I wanted find out if every subgroup of [itex]G \times H[/itex] is realised as a direct product of subgroups of G and H. Apparently it is not, because the diagonal subgroup in [itex]G \times G [/itex] disproves this. I'm a little confused, because I thought the proof I wrote was correct for a subgroup write [itex]A \times B [/itex] where A is a subset of G, and B a subset of H. Can't you show A is a subgroup of G using [itex] (g,1) [/itex] and analogously with B? For example m,n in A then [itex] (m,1),(n,1) [/itex] are in [itex]A \times B [/itex]. Hence [itex] (mn,1) [/itex] is and therefore mn is in A? There must be something wrong? Is the property true for certain type of groups? But I didn't use anything about G and H.