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QUICK PHRASES: no other isomorphic subgroups, no isomorphic copies, only subgroup of its isomorphism type
- Whenever such that , then (i.e. and are the same subgroup).
- is a co-Hopfian group, and whenever such that , then .
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This is a variation of characteristic subgroup|Find other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup
This article describes a property that arises as the conjunction of a subgroup property: isomorph-containing subgroup with a group property (itself viewed as a subgroup property): co-Hopfian group
View a complete list of such conjunctions
- The trivial subgroup is isomorph-free.
- Any co-Hopfian group (and in particular, any finite group) is isomorph-free as a subgroup of itself.
Examples in small finite groups
Below are some examples of a proper nontrivial subgroup that satisfy the property isomorph-free subgroup.
Below are some examples of a proper nontrivial subgroup that does not satisfy the property isomorph-free subgroup.
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|normal Sylow subgroup||Sylow subgroup that is also normal||Follows from Sylow implies order-conjugate||Finite isomorph-free subgroup, Order-unique subgroup|FULL LIST, MORE INFO|
|normal Hall subgroup||Hall subgroup (i.e., order and index are relatively prime) that is also normal||Finite isomorph-free subgroup, Order-unique subgroup|FULL LIST, MORE INFO|
|order-unique subgroup||unique subgroup of its order||isomorph-free not implies order-unique (see also list of examples)|||FULL LIST, MORE INFO|
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
Here is a summary:
|Metaproperty name||Satisfied?||Proof||Difficulty level (0-5)||Statement with symbols|
|transitive subgroup property||No||isomorph-freeness is not transitive||It is possible to have groups such that is isomorph-free in and is isomorph-free in but is not isomorph-free in .|
|transitive subgroup property||Yes||isomorph-freeness is quotient-transitive||Suppose are groups such that is isomorph-free in and the quotient group is isomorph-free in . Then, is isomorph-free in .|
|intermediate subgroup condition||Yes||isomorph-freeness satisfies intermediate subgroup condition||Suppose are groups such that is isomorph-free in . Then, is also isomorph-free in .|
|finite-intersection-closed subgroup property||No||isomorph-freeness is not finite-intersection-closed||It is possible to have a group and subgroups of such that and are both isomorph-free but the intersection is not isomorph-free.|
|strongly join-closed subgroup property||Yes||isomorph-freeness is strongly join-closed||Suppose are subgroups of a group such that each is an isomorph-free subgroup of . Then, the join of subgroups is also an isomorph-free subgroup of .|
|finite-upper join-closed subgroup property||No||isomorph-freeness is not finite-upper join-closed||It is possible to have groups and are intermediate subgroups such that is isomorph-free in both and , but is not isomorph-free in .|
|trivially true subgroup property||Yes||The trivial subgroup is isomorph-free in any group.|
|identity-true subgroup property||No||It is possible for a group to be isomorphic to a subgroup of itself. The simplest example is , the group of integers, that is isomorphic to the subgroup for any positive integer . A group that is isomorph-free as a subgroup of itself is termed a co-Hopdian group.|
Effect of property operators
The subordination operator
A subgroup of a group is termed sub-isomorph-free if there is a series of subgroups , with each an isomorph-free subgroup of .
One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.GAP-codable subgroup property
View the GAP code for testing this subgroup property at: IsIsomorphFreeSubgroup
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
While there is no in-built command for testing whether a subgroup is isomorph-free, a short piece of GAP code can do the test. The code can be found at GAP:IsIsomorphFreeSubgroup, and the command is invoked as follows: