Base diagonal of a wreath product
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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]
The base diagonal of a wreath product is the subgroup comprising those members of the base power where all coordinates are equal. In the language of functions, it is the subgroup comprising the constant functions.
A subgroup is termed a base diagonal of a wreath product if it occurs as the base diagonal for some way of expressing the group as an internal wreath product.
Relation with other properties
- If is the base diagonal of a wreath product in a group , and is a nilpotent group of nilpotence class , then is also a subnormal subgroup of subnormal depth . In particular, if is abelian, then is normal. In fact, the following stronger statement is true: if has nilpotence class , then , with written times, is the trivial subgroup.