good day! i need to prove that the alternating group A(adsbygoogle = window.adsbygoogle || []).push({}); _{n}is a normal subgroup of symmetric group, S_{n}, 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 τεS_{n}, τA_{n}τ^{-1}is subset of A_{n}.

Let λετA_{n}τ^{-1}, then λ=τστ^{-1}, for all σεA_{n}. but since multiplication of transpositions are commutative and therefore,

λ=τστ^{-1}=σττ^{-1}=σ, thus, λεA_{n}, and therefore τA_{n}τ^{-1}is a subset of A_{n}.

II. I need to prove that A_{n}is a subset of τA_{n}τ^{-1}.

Let σεA_{n}and τεS_{n}, 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, A_{n}is a subset of τA_{n}τ^{-1}.

since I've shown that the two sets are subsets of each other, I therefore conclude that τA_{n}τ^{-1}=A_{n}and A_{n}is normal.

thanks and God bless!

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# Normal groups of permutation

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