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GroupTheory

 IsNormal
 test whether one group is contained as a normal subgroup of another

 Calling Sequence IsNormal( H, G )

Parameters

 H - a group G - a group

Description

 • A group $H$ is a normal subgroup of a group $G$ if $H$ is a subgroup of $G$, and if it is equal to each of its conjugates: $H={H}^{g}$, for all $g$ in $G$.
 • The IsNormal( H, G ) command tests whether the group H is a normal subgroup of the group G.  It returns true if H is normal in G, and returns false otherwise.  For some pairs H and G of groups, the value FAIL may be returned if IsNormal cannot determine whether H is a normal subgroup of G.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Symm}\left(4\right)$
 ${G}{≔}{{\mathbf{S}}}_{{4}}$ (1)
 > $H≔\mathrm{Alt}\left(4\right)$
 ${H}{≔}{{\mathbf{A}}}_{{4}}$ (2)
 > $\mathrm{IsNormal}\left(H,G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsNormal}\left(\mathrm{Alt}\left(5\right),G\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{IsSubgroup}\left(\mathrm{Alt}\left(5\right),G\right)$
 ${\mathrm{false}}$ (5)
 > $G≔\mathrm{Symm}\left(5\right)$
 ${G}{≔}{{\mathbf{S}}}_{{5}}$ (6)
 > $H≔\mathrm{DihedralGroup}\left(5\right)$
 ${H}{≔}{{\mathrm{D}}}_{{5}}$ (7)
 > $\mathrm{IsNormal}\left(H,G\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{IsSubgroup}\left(H,G\right)$
 ${\mathrm{true}}$ (9)

Compatibility

 • The GroupTheory[IsNormal] command was introduced in Maple 17.