Proving intermediate subgroup condition

A subgroup property $$p$$ is said to satisfy the intermediate subgroup condition if, whenever $$H \le K \le G$$ are groups and $$H$$ satisfies $$p$$ in $$G$$, $$H$$ also satisfies $$p$$ in $$K$$.

This article discusses methods to prove that a given subgroup property satisfies the intermediate subgroup condition. For methods to prove that a given subgroup property does not satisfy the intermediate subgroup condition, refer disproving intermediate subgroup condition.

Also refer:



Left-inner and left-extensibility-stable
Suppose $$a,b$$ are properties of functions from a group to itself. The subgroup property $$a \to b$$ is defined as the following property: a subgroup $$H$$ of a group $$G$$ satisfies property $$a \to b$$ in $$G$$ if, for any function from $$G$$ to itself satisfying property $$a$$, the restriction to $$H$$ satisfies property $$b$$. Such an expression for a subgroup property is termed a function restriction expression and a subgroup property that can be expressed in this fashion is termed a function restriction-expressible subgroup property.

We say that a subgroup property is a:


 * left-extensibility-stable subgroup property if it has a function restriction expression where the left side is an extensibility-stable function property: any function satisfying the property on a subgroup can be extended to a function satisfying the property on the whole group.
 * left-inner subgroup property if it has a function restriction expression where the left side is inner automorphisms. Note that the property of being an inner automorphism is extensibility-stable, so left-inner subgroup properties are left-extensibility-stable.

It turns out that any left-extensibility-stable subgroup property, and hence, any left-inner subgroup property satisfies the intermediate subgroup condition. Here are some examples:


 * Normality satisfies intermediate subgroup condition: Normality is a left-inner subgroup property -- it is in fact the invariance property with respect to inner automorphisms. Hence, it satisfies the intermediate subgroup condition.
 * Central factor satisfies intermediate subgroup condition
 * Transitive normality satisfies intermediate subgroup condition
 * Conjugacy-closed normality satisfies intermediate subgroup condition

Property of being a "factor"
Many subgroup properties that arise as being a factor with respect to some product notion satisfy the intermediate subgroup condition. Specifically, if $$H$$ is a normal subgroup such that $$H$$ is a factor of $$G$$ in some sense with some complementary factor, then for an intermediate subgroup $$K$$, we can find a smaller complementary factor, usually by intersecting with or restricting to $$K$$. Here are some examples:


 * Direct factor satisfies intermediate subgroup condition
 * Complemented normal satisfies intermediate subgroup condition
 * Permutably complemented satisfies intermediate subgroup condition
 * Central factor satisfies intermediate subgroup condition
 * Retract satisfies intermediate subgroup condition

Conjunction
The conjunction (AND) of two subgroup properties satisfying the intermediate subgroup condition also satisfies the intermediate subgroup condition.

Also, the conjunction of a subgroup property satisfying the intermediate subgroup condition and a group property interpreted as a subgroup property also satisfies intermediate subgroup condition.

Here are some examples: normal Sylow subgroup, normal Hall subgroup, abelian normal subgroup.

Disjunction
The disjunction (OR) of two subgroup properties satisfying the intermediate subgroup condition also satisfies the intermediate subgroup condition.

Also, the disjunction of a subgroup property satisfying the intermediate subgroup condition and a group property interpreted as a subgroup property also satisfies intermediate subgroup condition.

Intermediately operator
The intermediately operator takes as input a subgroup property and outputs the weakest subgroup property satisfying intermediate subgroup condition that is stronger than it. By definition, this operator always gives subgroup properties satisfying the intermediate subgroup condition. Here are some examples: intermediately characteristic subgroup, intermediately fully invariant subgroup, intermediately normal-to-characteristic subgroup, intermediately subnormal-to-normal subgroup.

Transfer condition operator
The transfer condition operator always outputs a subgroup property satisfying the transfer condition, and hence, also the intermediate subgroup condition. Here are some examples: transfer-closed characteristic subgroup, transfer-closed fully invariant subgroup, transfer-closed normal-to-characteristic subgroup, transfer-closed subnormal-to-normal subgroup.