Base diagonal of a wreath product

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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

• Diagonal subgroup of a direct power
• Direct factor

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.