Hi, Mateus. I'm not quite understanding your question. When we use the word "isomorphic", we are relating two groups to each other, as in "G is isomorphic to H". Asking if A4 is isomorphic begs the question: Isomorphic to what?
@@mateusbueno9535 It still isn't clear to me that what you are asking is what you want, but in case it is, the answer is yes. However, that has nothing to do with A4. Every group is isomorphic to itself. To see this, you go to the definition of isomorphic: G and H are isomorphic if there exists a bijective homomorphism from G to H. In our case, G = H, and we can use the identity map.
vishali Kalra A subgroup being abelian and being normal have little to do with each other. Let G be a group and let H be a subgroup of G. There are many examples where H is abelian and not normal in G (e.g., every subgroup of order 2 in S3) and many examples where H is non-abelian and normal in G (e.g., An in Sn for n > 3). Recall that H is normal in G if and only if H is closed under G-conjugation, i.e., ghg^-1 is an element of H for all g in G and h in H. You should observe that H being abelian (an internal property of H) tells you nothing about how elements of H interact with elements of G that aren’t in H. In short, normality is a relative property, not an absolute property. Calling a subgroup normal is similar to calling a person tall. It doesn’t make sense to call somebody tall unless you know what you’re comparing them to. I'm tall compared to a baby, but short compared to a mountain. Calling a subgroup abelian is like saying that somebody has five fingers on their left hand; you don’t need to compare them with anybody else to confirm that fact.
if a group is abelian then their subgroups are normal, converse is not true, as you noticed that A4 is not abelian but it is normal subgroup, also a group is normal if there is a unique(single) subgroup of that order. Here A4 is unique subgroup of S4 so it is normal.
