Abstract Algebra: For the following cyclic groups, list all generators and find the order of each group

a. The subgroup of S_6 generated by the permutation σ = (top row: 1 2 3 4 5 6, bottom row: 2 4 1 3 6 5)

b. The subgroup of GL (2, ℝ) generated by the matrix
[pi 0 ]
[0 (1/pi)]

a) In cycle notation, σ = (1 2 4 3)(5 6).
This has order lcm(4, 2) = 4.

So, this is cyclic of order 4 with generators σ and σ^3 = σ^(-1).
[The exponents for generators (given a fixed generator) should be relatively prime to the (finite) order of the cyclic group.]
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b) This is of infinite order, because no positive integral power of this matrix equals I.
As such, the only generators are the matrix itself and its inverse.

I hope this helps!