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

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 $d$>