Finite and any two maximal subgroups intersect trivially implies not simple non-abelian
From Groupprops
Contents
Statement
Suppose is a finite group with the property that any two distinct maximal subgroups of
intersect trivially. Then,
is not a simple non-abelian group.
Related facts
Direct applications
- Finite non-abelian and every proper subgroup is abelian implies not simple
- Finite non-nilpotent and every proper subgroup is nilpotent implies not simple
Indirect applications
- Classification of cyclicity-forcing numbers
- Classification of abelianness-forcing numbers
- Schmidt-Iwasawa theorem
Facts used
- The trivial subgroup is maximal if and only if the group is a group of prime order, which is a simple abelian group.
- Lagrange's theorem
- Group acts as automorphisms by conjugation: Thus, conjugates of a maximal subgroup are maximal and all have the same order.
- Size of conjugacy class of subgroups equals index of normalizer
- In a finite non-cyclic group, every element is contained in a maximal subgroup. This basically follows from the fact that cyclic iff not a union of proper subgroups.
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
We prove the statement by contradiction.
Given: A finite simple non-abelian group of order
such that any two distinct maximal subgroups of
intersect trivially.
To prove: A contradiction.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Every maximal subgroup of ![]() |
Fact (1) | ![]() |
[SHOW MORE] | |
2 | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Facts (2), (3), (4) | ![]() ![]() Any two maximal subgroups of ![]() |
Step (1) | [SHOW MORE] |
3 | The union of conjugates of any one maximal subgroup has at least ![]() |
Step (2) | [SHOW MORE] | ||
4 | We have the desired contradiction, because cardinality considerations force ![]() |
Fact (5) | Any two maximal subgroups of ![]() |
Step (3) | [SHOW MORE] |