I am confused between the usage of two words for all and any. Let us consider the example of the definition normal subgroups, A subgroup $H$ is said to be normal if $\forall g \in G, g^{-1}Hg = H$ but if I rephrase the definition of normal subgroup to $H$ is normal in $G$ if for any $g \in G, g^{-1}Hg = H$.
Are these two definitions correct?
On
That is how the word "any" is commonly used in mathematics. However, in many contexts, especially outside mathematics, "any" means "some". For instance, "If you are carrying any items on the following list, you must declare them." I usually avoid the word "any" in mathematics because of such potential ambiguity.
Yeap, if something is true "for any" that means it does not matter which of them you take, which means it is true "for all".