good day! i need to prove that the alternating group An is a normal subgroup of symmetric group, Sn, and i just want to know if my proving is correct. we know that normal subgroup is subgroup where the right and left cosets coincides. but i got this equivalent definition of normal group from fraleigh's book, which states that for all gεG and hεH, a subgroup H of G is normal iff gHg-1=H. now here's my proof using the definition i got, I. I need to show that for all τεSn, τAnτ-1 is subset of An. Let λετAnτ-1, then λ=τστ-1, for all σεAn. but since multiplication of transpositions are commutative and therefore, λ=τστ-1=σττ-1=σ, thus, λεAn, and therefore τAnτ-1 is a subset of An. II. I need to prove that An is a subset of τAnτ-1. Let σεAn and τεSn, since σ is an even transposition, τσ must be an odd transposition since no permutation is a product of both even or odd transposition. Also, since multiplication of transposition is commutative I now have, σ=σe=σττ-1=τστ-1 thus, An is a subset of τAnτ-1. since I've shown that the two sets are subsets of each other, I therefore conclude that τAnτ-1=An and An is normal. thanks and God bless!