Amalgam-characteristic subgroup
From Groupprops
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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]
Definition
Definition with symbols
Suppose is a group and
is a subgroup of
. We say that
is an amalgam-characteristic subgroup of
if
is characteristic in the group
given by:
.
In other words, is the amalgam of
with itself over
, and
is treated as the subgroup of
given by the amalgamated
between the two factors.
Relation with other properties
Stronger properties
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
normal subgroup | amalgam-characteristic implies normal | normal not implies amalgam-characteristic | |FULL LIST, MORE INFO |
Incomparable properties
- Characteristic subgroup: For full proof, refer: Characteristic not implies amalgam-characteristic Note that the other non-implication is clear from, for instance, finite normal implies amalgam-characteristic, because not every finite normal subgroup is characteristic (For full proof, refer: Normal not implies characteristic).