Some quicker methods to decide for or against normality .

Question

When testing for normality of a group apart from using the method of using generating sets E.G. If A is a generating set for G and B is a generating set for H then we must check if $bab^{-1}, b^{-1}ab$, (which cuts the work down a little but is still quite tediuous. ).

Is there some quick ways to decide for or against normality . I know a few , for instance in a nilpotent group the maximal subgroups are always normal, or groups of index two are always normal. I also know that in general this is quite an intractable question but I was hoping someone might know of some quick ways to decide normality that I'm unaware of. Particularly in deciding if a group is not normal as I know no tricks at all for deciding that.

Answer 1

If a subgroup is normal then it must be a (disjoint) union of $G$-conjugacy classes (the converse is not true). So if conjugacy classes are known (not hard to find them), then you can check against them.

Answer 2

To prove that a subgroup $H < G$ is not normal, you simply have to find one element $g \in G$ such that $gHg^{-1} \ne H$. It often happens that through intuition or experience or whatever, you can guess the right $g$, and then prove the inequality $g H g^{-1} \ne H$ quite easily.

In fact, sometimes normality fails so spectacularly that any element $g \in G-H$ satisfies $g H g^{-1} \ne H$. Such subgroups are called malnormal. As an example, in a rank 2 free group $F$ with free basis $a,b$, the subgroup generated by $a$ is malnormal.