nLab
wreath product of groups

Definition

Let HSym(X)H\leq Sym(X) and GSym(Y)G\leq Sym(Y) be permutation groups. Then their wreath product HGH \wr G is defined as the semidirect product H YGH^Y \rtimes G where GG operates on H YH^Y by permuting the components. Note that this is itself a permutation group, acting on X×YX\times Y by letting GG act trivially on the first and naturally on the second factor and letting H YH^Y act on X×YX\times Y such that the yy-th component of H YH^Y permutes X×{y}X\times\{y\} naturally and fixes everything else pointwise.

Examples

  • Every group GG is a permutation group on its underlying set |G||G| (the regular representation aka the Cayley-Yoneda embedding GSym(|G|)G\hookrightarrow Sym(|G|)) and often only this special case is considered so that people speak of “the wreath product HGH \wr G” for abstract groups GG and HH without mentioning their permutation representations.
  • Dih(4)Dih(4) the dihedral group of order 24=82\cdot 4=8 is the wreath product C 2C 2C_2 \wr C_2 (in this sense of regular representations).
  • The sylow-pp-subgroups P nP_n of the symmetric groups Sym(n)Sym(n) can be recursively described as the wreath product C pP aC_p \wr P_a where C pC_p is the cyclic group of order pp and n=ap+rn=ap+r with 0rp0\leq r \lneq p.
  • The sylow-\ell-subgroups of GL n(q)GL_n(q) for gcd(q,)=1gcd(q,\ell)=1 can be recursively described as a wreath product of the sylow-\ell-subgroup of a strictly smaller GL m(q)GL_{m}(q) and a sylow-\ell-subgroups of a suitable symmetric group.

Relation to the Borel construction

The homotopy quotient described in the Borel construction,

(X//G) isoX× G(EG) , (X //G)_\bullet \simeq_{iso} X \times_G (E G)_\bullet \,,

has the wreath product π 1(X)G\pi_1(X)\wr G for fundamental group.

(…)

References

(…)

Last revised on November 3, 2017 at 02:55:06. See the history of this page for a list of all contributions to it.