Question:Characteristic subgroup other isomorphic subgroups
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Q: If is a characteristic subgroup of , does that mean there is no other subgroup of isomorphic to ?
A: Not in general.
Call a subgroup of a group an isomorph-free subgroup if there is no other subgroup of the group isomorphic to it. Then, isomorph-free implies characteristic but characteristic not implies isomorph-free. In fact:
- Call a subgroup a normal-isomorph-free subgroup if it is normal and there is no other normal subgroup isomorphic to it. Then characteristic not implies normal-isomorph-free.
- In a similar vein, characteristic not implies series-isomorph-free, characteristic not implies quotient-isomorph-free, and characteristic not implies characteristic-isomorph-free.
Basically, characteristic just means invariant under automorphisms of the whole group. Being the only subgroup of its type in any of the above senses is good enough to force characteristicity, but a subgroup could be characteristic for many other reasons.