Isomorph-free subgroup
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
Definition
QUICK PHRASES: no other isomorphic subgroups, no isomorphic copies, only subgroup of its isomorphism type
A subgroup of a group
is said to be isomorph-free if it satisfies the following equivalent conditions:
- 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
Examples
Extreme examples
- 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
Stronger 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 |
Weaker properties
Metaproperties
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
transitive subgroup property | Yes | isomorph-freeness is quotient-transitive | Suppose ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
intermediate subgroup condition | Yes | isomorph-freeness satisfies intermediate subgroup condition | Suppose ![]() ![]() ![]() ![]() ![]() | |
finite-intersection-closed subgroup property | No | isomorph-freeness is not finite-intersection-closed | It is possible to have a group ![]() ![]() ![]() ![]() ![]() ![]() | |
strongly join-closed subgroup property | Yes | isomorph-freeness is strongly join-closed | Suppose ![]() ![]() ![]() ![]() ![]() ![]() | |
finite-upper join-closed subgroup property | No | isomorph-freeness is not finite-upper join-closed | It is possible to have groups ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
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 ![]() ![]() ![]() |
Effect of property operators
The subordination operator
Applying the subordination operator to this property gives: sub-isomorph-free subgroup
A subgroup of a group
is termed sub-isomorph-free if there is a series of subgroups
, with each
an isomorph-free subgroup of
.
Testing
GAP code
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:
IsIsomorphFreeSubgroup(group,subgroup);