Profinite groups where centralizers of nth powers are finite or open have an open normal subgroup T making G/Z(T) of finite exponent.
I.: Groups and Lie rings admitting almost regular automorphisms of prime order,Rend
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Profinite groups with restricted centralizers of powers
Profinite groups where centralizers of nth powers are finite or open have an open normal subgroup T making G/Z(T) of finite exponent.