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Abstract/Modern Algebra.

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Added on  2020-05-08

Abstract/Modern Algebra.

   Added on 2020-05-08

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Abstract/Modern AlgebraQuestion:Let G be cyclic and of prime order p. How many automorphisms does G x G have?SolutionAssuming G is group and¿G¿p, a primeConsidering Y=G×G is a space vector κ=p having an action α(x,y) that is (αx,αy); thus for any group homomorphism is given by:m:YY is a κ- linear which lies in GL(Y)=GL2(p). Therefore, Aut(G)=GL2(p). Determining the order of the latter is given as follows:g=(ijkl) have a determinant det(g)=iljkConsidering a two case scenario:1)Assuming i,l0 and il=jk that means det(g)=0: for this scenario, there are (p1)3pairs whichis (i,j,k,l)=(x1,x2,x3,x2x3x1) for x1,x2,x3p×2)If il=0jK=0; that is il=0jk=0 resulting to det(g)=0: for this scenario, there are(2p1)2 pair. From the two (20 scenario, there are a total of p4 matrices gHence, there are p4(p1)3(2p1)2Factorizing out p4(p1)3(2p1)2 gives:(p21)(p2p)Thus, there are:(p21)(p2p)automorphisms
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