Subgroups of \(p\)-groups

Algebra Level pending

Let \(G\) be a group of order \(p^5,\) where \(p\) is prime. Consider the following statements:

I. \(G\) has a normal subgroup of order \(p.\)
II. \(G\) has a normal subgroup of order \(p^2.\)
III. \(G\) has a normal subgroup of order \(p^3.\)
IV. \(G\) has a normal subgroup of order \(p^4.\)

How many of the statements I-IV is/are always true for any \(G\) of order \(p^5\)?

Terminology: The order of a finite group is the number of elements in the group.

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