Abstract/Modern Algebra.

Added on - 08 May 2020

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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 primeConsideringY=G×Gis a space vectorκ=phaving an actionα(x,y)that is(αx,αy); thus for anygroup homomorphism is given by:m:YYis aκ- linear which lies inGL(Y)=GL2(p). Therefore,Aut(G)=GL2(p).Determining the order of the latter is given as follows:g=(ijkl)have a determinantdet(g)=iljkConsidering a two case scenario:1)Assumingi,l0andil=jkthat meansdet(g)=0: for this scenario, there are(p1)3pairs whichis(i,j,k,l)=(x1,x2,x3,x2x3x1)forx1,x2,x3p×2)Ifil=0jK=0; that isil=0jk=0resulting todet(g)=0: for this scenario, there are(2p1)2pair.From the two (20 scenario, there are a total ofp4matricesgHence, there arep4(p1)3(2p1)2Factorizing outp4(p1)3(2p1)2gives:(p21)(p2p)Thus, there are:(p21)(p2p)automorphisms
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