Nilpotent derived subgroup implies subnormal join property
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group with nilpotent derived subgroup) must also satisfy the second group property (i.e., group satisfying subnormal join property)
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- Nilpotent implies every subgroup is subnormal
- Join of subnormal subgroups is subnormal iff their commutator is subnormal
Given: A group such that is nilpotent. Subnormal subgroups of .
To prove: is subnormal.
Proof: Clearly, . Since is nilpotent by assumption, fact (1) tells us that is a subnormal subgroup of . Further, is a normal subgroup of , so is subnormal in . Thus, by fact (2), issubnormal in .