Isomorph-free subgroup

Symbol-free definition
A subgroup of a group is said to be isomorph-free if it satisfies the following equivalent conditions:


 * 1) There is no other subgroup of the group isomorphic to it as an abstract group.
 * 2) It is an defining ingredient::isomorph-containing subgroup that is also a defining ingredient::co-Hopfian group (in other words, it contains every subgroup isomorphic to it, but no proper subgroup of it is isomorphic to it).

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be isomorph-free if it satisfies the following equivalent conditions:


 * 1) Whenever $$K \le G$$ such that $$H \cong K$$, then $$H = K$$ (i.e. $$H$$ and $$K$$ are the same subgroup).
 * 2) $$H$$ is a co-Hopfian group, and whenever $$K \le G$$ such that $$H \cong K$$, then $$K \le H$$.

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.

Metaproperties
An isomorph-free subgroup of an isomorph-free subgroup need not be isomorph-free.

If $$H$$ is an isomorph-free subgroup of $$G$$ and $$K/H$$ is an isomorph-free subgroup of $$G/H$$, then $$K$$ is an isomorph-free subgroup of $$G$$.

If $$H, K$$ are isomorph-free subgroups of $$G$$, the intersection $$H \cap K$$ need not be isomorph-free.

If $$H_i, i \in I$$ is a collection of isomorph-free subgroups of $$G$$, the join of the $$H_i$$s is also isomorph-free.

If $$H \le G$$ and $$K,L$$ are intermediate subgroups such that $$H$$ is isomorph-free in both $$K$$ and $$L$$, $$H$$ need not be isomorph-free in $$\langle K, L \rangle$$.

Trimness
The property of being isomorph-free is trivially true, viz., it is satisfied by the trivial subgroup. However, a group need not be isomorph-free in itself, because it may be isomorphic to a proper subgroup of itself (the condition of being isomorph-free as a subgroup of itself, is precisely the condition of being a co-Hopfian group).

Effect of property operators
A subgroup $$H$$ of a group $$G$$ is termed sub-isomorph-free if there is a series of subgroups $$H = H_0 \le H_1 \le \dots \le H_n = G$$, with each $$H_{i-1}$$ an isomorph-free subgroup of $$H_i$$.

Testing
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);