Equivalence of definitions of nilpotent variety
This article gives a proof/explanation of the equivalence of multiple definitions for the term nilpotent variety of groups
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The two definitions that we have to prove as equivalent
Consider a subvariety of the variety of groups, i.e., a collection of groups (closed upto isomorphism) that is closed under taking subgroups, quotients, and arbitrary direct products. Then, the following are equivalent:
- Every group in the variety is a nilpotent group
- There exists a constant such that every group in the variety is nilpotent with nilpotence class at most .
Clearly, (2) implies (1). We need to show that (1) implies (2). We do this by contradiction: suppose every group in the variety is nilpotent, and suppose that, for every , there exists a in the variety with nilpotence class strictly greater than . Then, let be the direct product of all the s.
Suppose the group is nilpotent of nilpotence class . Then, every subgroup of has nilpotence class at most . But, has a direct factor of nilpotence class more than , leading to a contradiction.