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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$A≔\left\{\mathrm{Perm}\left(\left[\left[1\,2\,4\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)\right\}$

$A≔\left\{\left(1\,2\,4\right)\,\left(1\,2\right)\left(3\,4\right)\right\}$

$B≔\left\{\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,3\,4\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,3\right]\,\left[2\,4\right]\right]\right)\right\}$

$B≔\left\{\left(1\,3\right)\left(2\,4\right)\,\left(1\,2\,3\right)\,\left(2\,3\,4\right)\right\}$

$C≔\mathrm{FrobeniusProduct}\left(A\,B\,G\right)$

$C≔\left\{\left(1\,4\,3\right)\,\left(1\,3\,4\right)\,\left(1\,4\right)\left(2\,3\right)\,\left(1\,3\,2\right)\,\left(1\,3\right)\left(2\,4\right)\right\}$

$\mathrm{nops}\left(C\right)\ne \mathrm{nops}\left(A\right)\mathrm{nops}\left(B\right)$

$5\ne 6$

$A≔\mathrm{ConjugacyClass}\left(\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)\,G\right)$

$A≔{\left(\left(1\,2\right)\left(3\,4\right)\right)}^{{\mathbf{A}}_4}$

$B≔\mathrm{ConjugacyClass}\left(\mathrm{Perm}\left(\left[\left[1\,2\,4\right]\right]\right)\,G\right)$

$B≔{\left(\left(1\,2\,4\right)\right)}^{{\mathbf{A}}_4}$

$C≔\mathrm{FrobeniusProduct}\left(\mathrm{Elements}\left(A\right)\,\mathrm{Elements}\left(B\right)\,G\right)$

$C≔\left\{\left(1\,4\,3\right)\,\left(1\,3\,2\right)\,\left(1\,2\,4\right)\,\left(2\,3\,4\right)\right\}$

$\mathrm{numelems}\left(C\right)$

$4$

$\mathrm{numelems}\left(A\right)\mathrm{numelems}\left(B\right)$

$12$

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

For more information on Maple 2015 changes, see Updates in Maple 2015.