AEP does not satisfy intermediate subgroup condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., AEP-subgroup) not satisfying a subgroup metaproperty (i.e., intermediate subgroup condition).
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Statement with symbols
It is possible to have groups such that is an AEP-subgroup of but is not an AEP-subgroup of .
Example of an Abelian group
Let and be isomorphic copies of . Let and be subgroups of order two in and respectively. Then, define:
We claim that:
- : This is clear from the definition.
- is an AEP-subgroup of
- is not an AEP-subgroup of : Consider the automorphism of that exchanges the generators of and . This cannot extend to an automorphism of , because in , the generator of is the double of an element, while the generator of is not the double of anything.