ComplexProduct - Maple Help

GroupTheory

 FrobeniusProduct
 compute the product of two complexes in a finite group

 Calling Sequence FrobeniusProduct( A, B, G ) ComplexProduct( A, B, G )

Parameters

 A - a subset of G B - a subset of G G - a group

Description

 • The Frobenius product of two complexes $A$ and $B$ in a group $G$ is the set of all elements $\mathrm{ab}$, with $a\in A$ and $b\in B$.
 • The  FrobeniusProduct( A, B, G ) command computes the product of the complexes A and B in the finite group G.
 • The ComplexProduct command is provided as an alias.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $A≔\left\{\mathrm{Perm}\left(\left[\left[1,2,4\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right)\right\}$
 ${A}{≔}\left\{\left({1}{,}{2}{,}{4}\right){,}\left({1}{,}{2}\right)\left({3}{,}{4}\right)\right\}$ (2)
 > $B≔\left\{\mathrm{Perm}\left(\left[\left[1,2,3\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,3,4\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,3\right],\left[2,4\right]\right]\right)\right\}$
 ${B}{≔}\left\{\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{2}{,}{3}\right){,}\left({2}{,}{3}{,}{4}\right)\right\}$ (3)
 > $C≔\mathrm{FrobeniusProduct}\left(A,B,G\right)$
 ${C}{≔}\left\{\left({1}{,}{4}{,}{3}\right){,}\left({1}{,}{3}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right){,}\left({1}{,}{3}{,}{2}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)\right\}$ (4)
 > $\mathrm{nops}\left(C\right)\ne \mathrm{nops}\left(A\right)\mathrm{nops}\left(B\right)$
 ${5}{\ne }{6}$ (5)
 > $A≔\mathrm{ConjugacyClass}\left(\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right),G\right)$
 ${A}{≔}{\left(\left({1}{,}{2}\right)\left({3}{,}{4}\right)\right)}^{{{\mathbf{A}}}_{{4}}}$ (6)
 > $B≔\mathrm{ConjugacyClass}\left(\mathrm{Perm}\left(\left[\left[1,2,4\right]\right]\right),G\right)$
 ${B}{≔}{\left(\left({1}{,}{2}{,}{4}\right)\right)}^{{{\mathbf{A}}}_{{4}}}$ (7)

Note that you must compute the elements of these conjugacy classes before computing their complex product.

 > $C≔\mathrm{FrobeniusProduct}\left(\mathrm{Elements}\left(A\right),\mathrm{Elements}\left(B\right),G\right)$
 ${C}{≔}\left\{\left({1}{,}{4}{,}{3}\right){,}\left({1}{,}{3}{,}{2}\right){,}\left({1}{,}{2}{,}{4}\right){,}\left({2}{,}{3}{,}{4}\right)\right\}$ (8)
 > $\mathrm{numelems}\left(C\right)$
 ${4}$ (9)
 > $\mathrm{numelems}\left(A\right)\mathrm{numelems}\left(B\right)$
 ${12}$ (10)

Compatibility

 • The GroupTheory[FrobeniusProduct] command was introduced in Maple 2015.