# Abstract Algebra-Pre Exam questions

1. Feb 9, 2010

### TheForumLord

1. The problem statement, all variables and given/known data

Question 1: Let $$\sigma_{1}, \sigma_{2}$$ be two cycles of length n in the symmetry group $$S_{n}$$. Prove that if $$\sigma_{1} & \sigma_{2}$$ commute then there is a natural number r such as $$\sigma_{1}^{r} = \sigma_{2}$$ .

Question 2:
A. Does the groups $$D_{3}xZ_{5}$$ and $$D_{5}xZ_{3}$$ are isomorphic?
Try to solve the question by checking the orders of elements in the center.
B. Give an example of two groups of order 40 which aren't abelian and both aren't isomorphic.