Supersolvable implies nilpotent derived subgroup
From Groupprops
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., supersolvable group) must also satisfy the second group property (i.e., group with nilpotent derived subgroup)
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Statement
The derived subgroup of a supersolvable group is a nilpotent group. Moreover, the nilpotency class of the derived subgroup is bounded from above by the length of any normal series for the whole group where each of the quotient groups between successive members is a cyclic group.
Facts used
- Derived subgroup centralizes cyclic normal subgroup
- Derived subgroup is normal
- Normality is strongly intersection-closed
- Normality satisfies image condition
- Second isomorphism theorem
- Cyclicity is subgroup-closed
- Derived subgroup satisfies image condition: Under a surjective homomorphism, the image of the derived subgroup equals the derived subgroup of the image.
Proof
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
Given: A supersolvable group with a normal series
with
cyclic.
To prove: is nilpotent of class at most
.
Proof: We will prove that the series:
is a central series for .
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() ![]() |
Fact (2) | |||
2 | ![]() ![]() ![]() |
Fact (3) | ![]() ![]() |
Step (1) | |
3 | ![]() ![]() ![]() |
Fact (4) | ![]() ![]() |
Step (2) (applied to both ![]() ![]() |
|
4 | ![]() |
Fact (6) | ![]() |
Step (2) (applied to ![]() |
By the second isomorphism theorem (fact (5)), this quotient is isomorphic to ![]() ![]() |
5 | ![]() ![]() |
Fact (7) | |||
6 | ![]() ![]() |
Fact (1) | Steps (3), (4), (5) | ![]() ![]() ![]() ![]() ![]() | |
7 | The series is indeed a central series. | Step (6) (in light of step (2)) |