by "The Kourovka Notebook. Unsolved Problems in Group Theory" there is a strong conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p.
What about finite abelian p-group?
Is there a finite abelian p-group without noninner automorphism of order p?
If yes can we for example by changing "order p" in conjecture make it for all finite p-group?
2026-03-26 17:34:27.1774546467
about conjecture "every finite nonabelian p-group admits a noninner automorphism of order p"
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For abelian $p$-groups this problem is not that hard.
If it has a cyclic group of order $p^2$ as a factor, then this contributes the needed automorphism.
If not then it is elementary abelian, and the order of the automorphism group (which is $GL_n(\mathbb{F}_p)$) is again divisible by $p$ as long as it is not cyclic of prime order.
Clearly if the group is cyclic of prime order then it has no automorphism of order $p$.