Wreath Product

It combines repeated copies with a group action. In A ≀ H, A is the group repeated across a set of positions, and H permutes those copies. The result is a particular semidirect product, not ordinary multiplication. The order matters: H supplies the action.

The construction did not come from one isolated invention. Historical accounts trace finite-group versions to Cauchy in 1844, Netto in 1882, and Radzig in 1895, in work involving Sylow subgroups—the largest subgroups of prime-power order—of symmetric groups, which are groups of permutations of finite sets.

In 1937, Pólya used the German name “Kranzgruppe” (“wreath group”), which led to the English name. The Krasner–Kaloujnine embedding theorem appeared in 1950 and 1951; by 1965, wreath products had also reached finite-semigroup and automata theory, where they help describe step-by-step machine decompositions.

Why do some books write ≀ differently?

The notation can mark different wreath product versions. For infinitely many positions, a restricted wreath product allows only finitely many non-identity entries among the copies; an unrestricted one allows arbitrary entries.

Authors may use different symbols or leave the index set unstated. Check the author's definition before treating two similar-looking formulas as the same construction.