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  "body": "\n## Definition\n\nLet $A$ and $G$ be [[groups|G5S1M9]], and let $G$ [[act|0WCY42EV9V]] on a set $\\Omega$, written $(g, \\omega) \\mapsto g\\omega$.\nLet $A^\\Omega$ be the group of all functions $f \\colon \\Omega \\to A$ with the pointwise product, that is, the [[direct product|00HF4RJ5QN]] of $\\abs{\\Omega}$ copies of $A$.\nFor $g \\in G$ and $f \\in A^\\Omega$, let $g \\cdot f \\in A^\\Omega$ be the function $(g \\cdot f)(\\omega) = f(g^{-1}\\omega)$.\nThis defines an action of $G$ on $A^\\Omega$ by automorphisms.\nThe *wreath product* $A \\wr_\\Omega G$ is the [[semidirect product|0H3WZXCZNF]] $A^\\Omega \\rtimes G$ for this action.\nIts elements are the pairs $(f, g)$ with $f \\in A^\\Omega$ and $g \\in G$, and the product is\n$$(f, g)(f', g') = (f \\, (g \\cdot f'), \\; g g').$$\nThe subgroup $A^\\Omega$ is the *base group*.\nWhen $\\Omega = G$ with the action by left multiplication, the group is the *regular wreath product*, written $A \\wr G$.\n\n## Remarks\n\nThe order of a finite wreath product is $\\abs{A \\wr_\\Omega G} = \\abs{A}^{\\abs{\\Omega}} \\abs{G}$.\n\nExample: let $A = C_2$ and let $G = C_2$ act on $\\Omega = \\set{1, 2}$ by swapping.\nThen $A \\wr_\\Omega G$ has order $2^2 \\cdot 2 = 8$.\nIt is isomorphic to the dihedral group of order $8$, the symmetry group of a square.\nThe base group $C_2 \\times C_2$ consists of the identity, the half-turn and the reflections in the two diagonals.\nThe swap is the reflection in an axis through the midpoints of two opposite sides.\n\nNon-example: the direct product $A^\\Omega \\times G$ has the same underlying set, but there $G$ does not move the functions at all.\nIts multiplication differs from that of the wreath product when $A \\ne 1$ and $G$ moves some point, because then $g \\cdot f \\ne f$ for some $g$ and $f$.\n\nFor $G = S_m$ acting on $\\set{1, \\dots, m}$, the group $S_k \\wr S_m$ is the subgroup of $S_{km}$ that preserves a fixed partition of $km$ points into $m$ parts of size $k$.\nThe group $C_2 \\wr S_n$ is the group of signed permutations, of order $2^n n!$.\nFor an infinite set $\\Omega$, the group defined above is the *unrestricted* wreath product, built from all functions $\\Omega \\to A$.\nThe *restricted* wreath product uses only the functions with $f(\\omega) = 1$ for all but finitely many $\\omega$.\n",
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