There are two answers I can give here: no and possibly. First, no: the direct product of two subgroups is never a subgroup. It's not even a subset! If G is a group and H and K are subgroups, then elements of HxK are ordered pairs (h,k), which are not the same type of element as in G. So HxK isn't even a subset of G. But this leads to the next question: Is it possible that HxK is _isomorphic_ to HK in G? The answer is possibly. It depends on H and K. When this happens, HK is called the "internal direct product" of H and K, since the direct product structure comes from putting the groups together using the operation inside of G, rather than putting the two groups together outside of G (like we do with the standard/external direct product). These concepts are addressed at the end of this playlist, in the videos titled "Internal and External Direct Products".
Thanks so much for this. It really helps to have videos like these while reading through D&F
Ohh... thank you... it's so practically explained
So by "composed" do you mean composed under the group operation? Like taking and in Z6 might mean = {0,3,2,5,4,1}= Z6?
Oh it's so hard
But the question is direct product of two subgroup also subgroup?
There are two answers I can give here: no and possibly.
First, no: the direct product of two subgroups is never a subgroup. It's not even a subset! If G is a group and H and K are subgroups, then elements of HxK are ordered pairs (h,k), which are not the same type of element as in G. So HxK isn't even a subset of G.
But this leads to the next question: Is it possible that HxK is _isomorphic_ to HK in G? The answer is possibly. It depends on H and K. When this happens, HK is called the "internal direct product" of H and K, since the direct product structure comes from putting the groups together using the operation inside of G, rather than putting the two groups together outside of G (like we do with the standard/external direct product).
These concepts are addressed at the end of this playlist, in the videos titled "Internal and External Direct Products".
The pudding is in the proof 🎉
The Theorem looks deceptively simple but the proof is very convulating. My logics can't comprehend it...
Part 2: th-cam.com/video/s-2PGXuHM68/w-d-xo.html
Part 3: th-cam.com/video/KIMMmUIBI0c/w-d-xo.html