SOME COMPUTATIONS YOU SHOULD BE ABLE TO DO

· Compute with complex numbers (including computation in U and U_n)

This includes solving equations of the form zⁿ = (whatever)

· Compute using modular arithmetic ( “+_s” for a real number “s”)

This includes computing in Z_n

· Work with (potential) isomorphisms between U_n and Z_n

· Compute in any cyclic group (including Z, U_n , Z_n )

what are the generators?

what are the subgroups?

what’s the order of any element?

· Compute in the Klein 4-group

· Compute in any matrix group

· Compute in any of our integer, rational, real, or complex infinite groups

· Compute with permutations

in both “regular notation” and in cycle notation

find inverses, powers

find the order of an element

find the orbits of an element

write the element as a product of transpositions

is the element even or odd?

· Compute in “geometric” groups / dihedral groups / groups of symmetries

· Compute the left regular (or right regular) representation of a group element.

· Compute the cosets of a subgroup

· Compute in a coset group

· Compute in direct product groups

· Find the units and the zero divisors in the ring Z_n