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Group theory Consider the group GL2(Zp) of invertible 2X2 matrices with entries in the field Zp, where p is...

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Group theory

Consider the groupGL2(Zp)of invertible 2X2 matrices with entries in the field Zp, wherep is an odd prime.

Zp is an abelian group under addition,the group of unites of Zp is Zpx, which is an abeliangroup under multiplication. We say (Zp, +, ·) is a field.

    1. Show that the subsetD2(Zp) ofdiagonal matrices inGL2(Zp)is an abelian subgroup of order (p -1)2.
    1. For A, B?GL2(Zp),show that A and B are in thesame rightD2(Zp)-cosetif and only if there are non-zero elements ?, µ? Zp such thatA can be obtained from B bymultiplying the first column by ? and the secondcolumn by µ.
    2. For A,B?GL2(Zp),show that A and B are in thesame leftD2(Zp)-cosetif and only if there are non-zero elements ?, µ?Zp such thatA can be obtained from B bymultiplying the first column by ? and the secondcolumn by µ.
    3. Find a matrix   A such that theleft and rightD2(Zp)cosets containing A are different, that is, sothatD2(Zp)A?AD2(Zp).
    4. Find a non-identity matrix B such that theleft and rightD2(Zp)cosets containing B are the same, that is, sothatD2(Zp)A=AD2(Zp).

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