Is the definition given by the GAP-manual equivalent to the one given in the site?

Question

Here

http://groupprops.subwiki.org/wiki/Normalizer_of_a_subset_of_a_group

the normalizer of a subset of a group is defined.

GAP gives the following description of the Function IsNormal :

39.3.6 IsNormal . IsNormal(G, U) (operation) returns true if the group G normalizes the group U and false otherwise. A group G normalizes a group U if and only if for every $g\in G$ and $u \in U$ the element $u^g$ is a member of U. Note that U need not be a subgroup of G. Example gap> IsNormal(g,u); false

1) What does u^g mean (g is a group, so normally does not have integer entries) ?

2) If U is a subgroup of G, does IsNormal(G,U) always correctly decide whether U is a normal subgroup of G ?

Answer 1

The operation is conjugation; $u^g = gug^{-1}$. The one of the definitions of a normal subgroup: $$N \trianglelefteq G \Leftrightarrow gNg^{-1} \subseteq N \Leftrightarrow gng^{-1} \in N\ \ \ \forall n \in N, g\in G$$

So as long as you verify that $U$ really is a subgroup, that test should tell you if it is normal or not. So you want to run something like

IsNormal(G,H) and IsSubgroup(G,H);

which outputs true if $H$ is a normal subgroup of $G$, and false otherwise.