Isologic groups with respect to fixed nilpotency class lower than theirs have equal nilpotency class

From Groupprops
Jump to: navigation, search

Statement

Suppose G_1 and G_2 are groups that are isologic groups with respect to the variety of groups of nilpotency class c. Then, the following are true:

  1. G_1 is a nilpotent group if and only if G_2 is a nilpotent group
  2. Suppose G_1 is a group of nilpotency class at most d, with c \le d. Then, G_2 also has nilpotency class at most d
  3. Suppose G_1 is a group of nilpotency class exactly d, with c < d. Then, G_2 also has nilpotency class exactly d>

Related facts