Diagonal-in-square operator
From Groupprops
Definition
The diagonal-in-square operator is an operator that takes as input a subgroup property and outputs a group property as follows.
Suppose is a subgroup property. The diagonal-in-square operator on
gives the property of being a group
such that, in the external direct product
, the diagonal subgroup
satisfies property
.
Examples
- Applying the diagonal-in-square operator to the property of being a characteristic subgroup gives the property of being thetrivial group.
- Applying the diagonal-in-square operator to the property of being a normal subgroup gives the property of being an abelian group.
- Applying the diagonal-in-square operator to the property of being a subnormal subgroup gives the property of being a nilpotent group.