By a $p$-group, we mean a group in which every element has order a power of $p$. It is well known that finite $p$-group has non-trivial center. But, an infinite $p$-group may have trivial center.
Question: What are the properties of finite $p$-group which does not depend on the finiteness of the group? (such as maximal subgroup is normal; normalizer of a subgroup is larger than subgroup, commutator subgroup is proper, etc.)
The answer may be long (and even endless), so it will be better only to list the properties, and leave their verifications to the reader.