Max-equivalent to normal and intermediate subgroup condition implies subnormal in finite
Suppose is a subgroup property satisfying the following two conditions:
- If is maximal among the proper subgroups of a group satisfying , then is a normal subgroup.
- satisfies the intermediate subgroup condition: If and has property in , then has property in .
These two proofs are essentially the same; one is presented in an inductive style, and the other is presented in terms of construction of the subnormal series.
Given: A subgroup of a group satisfying property . is max-equivalent to normality and satisfies the intermediate subgroup condition.
To prove: is subnormal in .
- Consider the collection of proper subgroups of containing and satisfying property . Note that is nonempty (it contains ) and finite, so it has maximal elements under inclusion. Let be such an element.
- Since is max-equivalent to normality, is normal in .
- Since satisfies the intermediate subgroup condition, satisfies in . Thus, by induction on the order, is subnormal in .
- Combining the previous two steps, we see that is subnormal in .