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