For G being each of the following groups, compute the centre Z(G): (a) S3 (b)Z/3Z x S3

Let T : R² → P2(R) and U : Rª –→ M2x2 (R) be lincar transformations. A student claims T cannot be invertible because dim(R²) + dim(P2(R)). If the student is correct, prove their claim. If the student is not correct, explain why and give an example to illustrate. Clearly state whether or not the student is correct as part of your solution.

Find the order of the element 2+ (6) in the group Z30/(6).

The "cyclic" transformation T is defined by T( v1, v2,v 3) = v( 2, v3, vi). What is T(T(v))? What is T3(v)? What is T100 (v)? Apply Ta hundred times to v.