Give an example of a gp G & a subgroup H st N(H)=/C(H). Is there any containing relation between N(H) & C(H)?

Here, gp means group, N(H)={'a' belongs to G:aHa-1=H}, where a-1 is the inverse of 'a', st means such that, =/ means not equal to, C(H)={'x' belongs to G: xh=hx for all 'h' in H}. (Problem from abstract algebra)

Give an example of a gp G %26amp; a subgroup H st N(H)=/C(H). Is there any containing relation between N(H) %26amp; C(H)?
I like this question. I used to love group theory, but it's been decades since I studied it. If I say anything stupid in this response, let me know--I'm a little rusty, so I may screw up.





Since we clearly have to look at non-abelian groups, let's look at the obvious ones.





N(H) is defined as the largest subgroup of G having H as a normal subgroup. N(A4)) = S4 where A4 =Alternating group on a set of 4 elements, and S4 is the symmetric group.





However, S4 isn't abelian; even A4 isn't. So C(A4) is rather small, and doesn't even contain A4 since:


(123)(134) = (124) while


(134)(123) = (234)


all these are in A4 per


http://en.wikipedia.org/wiki/Alternating...





So clearly C(A4) is much smaller than H(A4).





It's also very clear that C(H) must be a subset of N(H) since if xh=hx, then xhx^-1 =h for all H and therefore xHx^-1 = H and x is in C(H).





Hey, I think I've still got it!