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]

Definition

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

Stronger properties

Facts

  • If H is the base diagonal of a wreath product in a group G, and H is a nilpotent group of nilpotence class c, then H is also a subnormal subgroup of subnormal depth c. In particular, if H is abelian, then H is normal. In fact, the following stronger statement is true: if H has nilpotence class c, then [\dots [G,H],H]\dots,H], with H written c times, is the trivial subgroup.