5. Determine if the following sets along with the given operations form groups. If so, determine...

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Advance Math

5. Determine if the following sets along with the givenoperations form groups. If so, determine the identity element andwhether or not they are Abelian. If not, explain why.
(a) GL(n, Z) where ? is matrix multiplication. This is thecollection of all n × n nonsingular matrices with integralentries.
(b) Sym(X) where X is a nonempty set and f ? Sym(X) if and onlyif f : X ? X is bijective where ? is composition.
(c) Aff(1, R), where Aff(1, R) := {fa,b : R ? R : fa,b(x) = ax +b, a, b ? R, a 6= 0} and ? is composition. These are called theone-dimensional affine functions. What happens if we allow a =0?
(d) T := {z ? C : |z| = 1} where ? is complex multiplication. Wewill again encounter T in later sections.
(e) SL(2, Z) where A ? SL(2, Z) if and only if A is a 2 × 2matrix of integers for which det A = 1. What about SL(n, Z) where n? N?

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4.1 Ratings (854 Votes)

a with operation is not a group Take Let Now is inverse of A which do not belong to So is not a groupis identity in b with composition as multiplicationthen are bijective Then we can See Answer

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