Characteristicity is not finite direct power-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) not satisfying a subgroup metaproperty (i.e., finite direct power-closed subgroup property).
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Statement with symbols
- Full invariance is finite direct power-closed, hence, fully invariant implies finite direct power-closed characteristic
Notion of finite direct power-closed characteristic
A subgroup of a group is termed a finite direct power-closed characteristic subgroup if for any natural number , is characteristic in .
Further information: direct product of Z8 and Z2
Suppose , and is the subgroup of order four given by:
Note that this subgroup is characteristic but not fully invariant, and is a standard example of the fact that characteristic not implies fully invariant in finite abelian group.
Then, we claim that is not a characteristic subgroup inside . To see this, consider the endomorphism of that projects onto the coordinate, i.e., the map:
Now, consider the automorphism of defined as:
This is an endomorphism as it is a sum of endomorphisms; it is an automorphism because it has an inverse:
In full, the automorphism is:
Under this automorphism, the element:
gets mapped to the element:
which is not inside .