I need to show subgroups of nilpotents groups are nilpotent.
I can get started but cannot finish.
Suppose G is a nilpotent group. Let H be a proper subgroup of G. Let K be a proper subgroup of H. I must show K is a proper subgroup of .
I assume . I know I must get a contradiction here but I can't figure it out.
Any hints?
yes, the "normalizer condition" that you're trying to prove is equivalent to "nilpotent" only for finite groups. but the fact is that a subgroup of any nilpotent group is nilpotent and we prove
results about nilpotent groups using central series not the normalizer condition!
do you only know the definition of "finite nilpotent groups" or you're also familiar with lower and upper central series, which give the definition of "nilpotent" for general (finite or infinite) groups?
yes we can definitely use that.
i tried to go that route as well, but maybe i didn't think hard enough about it.
Suppose H is a proper subgroup of finite nilpotent group G. Let Q be a Sylow p-subgroup of H. Then we get Q is a subgroup of the Sylow p-subgroup P of G. We know P is normal in G, and is unique.
If can P have another subgroup of the same order?