Characteristic not implies characteristic-isomorph-free in finite
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., characteristic-isomorph-free subgroup)
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- Characteristic-isomorph-free not implies normal-isomorph-free
- Normal-isomorph-free not implies isomorph-free
- Characteristic equals characteristic-isomorph-free in finite abelian group
Example of the infinite cyclic group
Let be the infinite cyclic group: the group of integers under addition. Then, is a characteristic subgroup of for any , and all the are isomorphic for a nonnegative integer. Thus, there are distinct characteristic subgroups that are isomorphic.
Example of a finite solvable group
Further information: symmetric group:S3
- The subgroup equals the center of , hence is characteristic.
- The subgroup equals the derived subgroup of , hence is characteristic.
Thus, we have two distinct characteristic subgroups that are isomorphic.
Example of a finite nilpotent group
There are groups of order with characteristic maximal subgroups that are isomorphic. For instance, the group with GAP Group ID has three pairwise isomorphic characteristic subgroups (each with group ID ).