A group $H$ is a subgroup of a group $G$ if $H$ is contained in $G$ and the binary operation on $H$ is the restriction of the binary operation on $G$ to $H$.

The IsSubgroup( H, G ) command tests whether the group H is a subgroup of the group G.  It returns true if H is a subgroup of G, and returns false otherwise.  For some pairs H and G of groups, the value FAIL may be returned if IsSubgroup cannot determine whether H is a subgroup of G.

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

$G≔\mathrm{Symm}\left(4\right)$

$G≔{\mathbf{S}}_4$

$H≔\mathrm{Alt}\left(4\right)$

$H≔{\mathbf{A}}_4$

$\mathrm{IsSubgroup}\left(H\,G\right)$

$\mathrm{true}$

$\mathrm{IsSubgroup}\left(\mathrm{Alt}\left(5\right)\,G\right)$

$\mathrm{false}$

The GroupTheory[IsSubgroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.