Finite non-abelian and every proper subgroup is abelian implies not simple
From Groupprops
Contents
Statement
Suppose is a non-abelian finite group such that every proper subgroup of is an abelian group. Then, is not a simple group.
Related facts
Further facts about every proper subgroup being abelian
- Finite non-abelian and every proper subgroup is abelian implies metabelian
- Slender nilpotent and every proper subgroup is abelian implies Frattini-in-center
- Classification of finite non-abelian groups in which every proper subgroup is abelian
- Schmidt-Iwasawa theorem
Analogues
- The analogous statement for infinite groups is not true. In fact, Tarski has constructed infinite groups in which every proper nontrivial subgroup is cyclic of prime order.
Applications
- Classification of cyclicity-forcing numbers: This is a classification of all natural numbers such that every group of order is cyclic.
Facts similar to ideas used in the proof techniques
Facts used
- Center is normal
- Subgroup of index two is normal
- Finite and any two maximal subgroups intersect trivially implies not simple non-abelian
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 finite non-abelian group such that every proper subgroup of is abelian.
To prove: is not simple.
Proof: We assume that is simple, and derive a contradiction. Let be the number of elements of .
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | is centerless, i.e., its center is trivial | Fact (1) | is simple non-abelian | -- | [SHOW MORE] |
2 | Any two maximal subgroups of intersect trivially | every proper subgroup of is abelian | Step (1) | [SHOW MORE] | |
3 | We have the desired contradiction. | Fact (3) | is finite simple non-abelian | Step (2) | Follows from Fact (3). |
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 149, Exercise 53 of Section 4.5 (Sylow's theorem), (Hint for solution given)^{More info}