Re: Centralizers, Normalizers and the Center

I'm confused by the notation. Is C_H(G). Is it (1) the set of all elements in H which commute with all of G? or (2) Z(H) the centralizer of H? If (1) it is the intersection of Z(G) with H and therefore a subgroup of G. If (2) it is a subgroup of H and therefore again a subgroup of G.
Also what is meant by N_H(G)? Might this be the set of all elements of G that commute with all elements of H? Is this a subroup of G? I'm not sure whether it is or isn't. If H instead of being a subgroup were just one element of G then of course N(a) is a subgroup
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