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Nilpotency of fixed class is direct product-closed

Contents

Statement

Version in terms of fixed class bound

Suppose G_i, i \in I is a collection of groups indexed by an indexing set I. Suppose there is a positive integer c such that each G_i is a nilpotent group of nilpotency class at most c.

Then, the external direct product of the G_is is also a nilpotent group of nilpotency class at most c.

Version in terms of maximum class

Suppose G_i, i \in I is a collection of groups indexed by an indexing set I. If all the G_is are nilpotent groups and there is a common finite bound on their nilpotency class values, then the external direct product of the G_is is also a nilpotent group and its nilpotency class is the maximum of the nilpotency class values of all the G_is.

In particular, for two nilpotent groups G_1 and G_2 of nilpotency classes c_1,c_2 respectively, the nilpotency class of G_1 \times G_2 equals \max \{ c_1, c_2 \}.

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