# Isomorph-normal characteristic of WNSCDIN implies weakly closed

From Groupprops

## Statement

Suppose are groups. Suppose, further, that is an Isomorph-normal characteristic subgroup (?) of and is a WNSCDIN-subgroup (?) of . Then, is a Weakly closed subgroup (?) of relative to .

## Facts used

- Characteristic of normal implies normal
- WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

## Proof

**Given**: Groups , such that is characteristic in , every subgroup of isomorphic to is normal in , and is a WNSCDIN-subgroup of .

**To prove**: is weakly closed in .

**Proof**:

- is normal in : Since is characteristic in and is normal in , fact (1) yields that is normal in .
- is normal in every conjugate of containing it: Suppose for some . Then, . Clearly, is isomorphic to . So, by the assumption, is normal in . Conjugating back, we get that is normal in .
- is weakly closed in with respect to : This follows from fact (2), using the previous two steps.