Automorphism of finite 2-group of Mersenne prime order acts nontrivially on Omega-1

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Statement

Suppose P is a finite 2-group, and \sigma is a non-identity automorphism of P such that the order of \sigma is a Mersenne prime greater than 3. Then, \sigma acts nontrivially on the subgroup \Omega_1(P) (the first omega subgroup).

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