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Let $G$ be a finite $p$-group. Suppose that $G$ has only one characteristic subgroup(nontrivial proper). What possible structures does $G$ have?

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Have a look at :

S. P. Glasby, P. P. Pálfy, Csaba Schneider, p-Groups with a unique proper non-trivial characteristic subgroup, Journal of Algebra, Volume 348,Pages 85-109

http://www.sciencedirect.com/science/article/pii/S0021869311005552

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