cyclic groups (2)

3 more problems describing some general properties enjoyed by cyclic groups

Superincreasing Sequence; Prove that Expression is Prime

Please assist me with the attached problems including: a Superincreasing Sequence and Proving that an Expression is Prime ... Thanks!

A problem related to period(order) of an element in a group

Let a and x be elements in a group G. Prove that a and axb ,where b is the inverse of a, have the same period.

A problem related to subgroups in group theory

Let H and K be subgroups of a group G such that one is not contained in the other. Prove that H U K is not a subgroup of G.

Irreducible and one-dimensional representations

1. Classify Irreducible representations of Z over C. 2. Classify one-dimensional representations of Sn over any field k such that char k is not equal to 2.