Endo-invariance implies strongly join-closed

Verbal statement
Any subgroup property that arises as an invariance property with respect to endomorphisms in the function restriction formalism is strongly join-closed, viz it is both join-closed and trivially true.

Symbolic statement
Let $$p$$ be an endomorphism property. Let $$I$$ be a (possibly empty) indexing set. Let $$H_i$$ is a family of subgroups of $$G$$ indexed by $$I$$. Assume that for every function $$f$$ on $$G$$ satisfying $$p$$, $$f(H_i)$$ &sube; $$H_i$$ (viz $$H_i$$ satisfies the invariance property for $$p$$).

Then, if $$H$$ denotes the join of (viz, subgroup generated by) all $$H_i$$s, $$H$$ also satisfies the invariance property for $$p$$. In other words, whenever $$f$$ is a function on $$G$$ satisfying $$p$$, $$f(H)$$ &sube; $$H$$.

Examples

 * A join of normal subgroups is normal. Here, the subgroup property of being normal is the property of being invariant under inner automorphisms, and is hence the invariance property for the property of being an inner automorphism.
 * A join of characteristic subgroups is characteristic. Here the subgroup property of being characteristic is the invariance property with respect to the property of being an automorphism.

Strongly join-closed subgroup property
A subgroup property is termed strongly join-closed if given any family of subgroups having the property, their join (viz the subgroup generated by them) also has the property. Note that just saying that a subgroup property is join-closed simply means that given any nonempty family of subgroups with the property, the join also has the property.

Thus, the property of being strongly intersection-closed is the conjunction of the properties of being intersection-closed and trivially true, viz satisfied by the trivial subgroup.

Facts used

 * 1) uses::Homomorphisms commute with joins: This states that if $$\sigma:G \to K$$ is a homomorphism, and $$H_i, i \in I$$ is a collection of subgroups of $$G$$ whose join is $$H$$, then $$\sigma(H)$$ is the join of $$\sigma(H_i), i \in I$$.

Proof
Given: A property $$p$$ of endomorphisms, a group $$G$$ with subgroups $$H_i, i \in I$$ whose join is a subgroup $$H$$ of $$G$$. Further, each $$H_i$$ is invariant under each endomorphism $$\sigma$$ of $$G$$ that satisfies $$p$$.

To prove: $$H$$ is invariant under each endomorphism $$\sigma$$ of $$G$$ that satisfies property $$p$$.

Proof: We pick any endomorphism $$\sigma$$ of $$G$$ that satisfies $$p$$.


 * 1) For each $$i \in I$$, $$\sigma(H_i)$$ is contained in $$H_i$$: This follows from the assumption on the $$H_i$$s.
 * 2) $$\sigma(H)$$ is the join of the $$\sigma(H_i)$$s: This follows from fact (1), with $$K = G$$.
 * 3) $$\sigma(H)$$ is contained in $$H$$: Since each $$\sigma(H_i)$$ is contained in $$H_i$$, it is in particular contained in $$H$$, hence their join, which is $$\sigma(H)$$, is also contained in $$H$$.

This completes the proof.