Abstract Algebra-Pre Exam questions

[TheForumLord]

Homework Statement

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.


Thanks in advance

Homework Equations

The Attempt at a Solution

I really have no idea in these questions...Help is needed asap... THanks!

 

[mighty2000]

For your first question, let us start by defining what it means for two cycles to commute. Two cycles \sigma_{1} and \sigma_{2} are said to commute if \sigma_{1} \sigma_{2} = \sigma_{2} \sigma_{1}. In other words, if applying \sigma_{1} and then \sigma_{2} gives the same result as applying \sigma_{2} and then \sigma_{1}.

Now, let us consider the case where \sigma_{1} and \sigma_{2} are both cycles of length n in the symmetry group S_{n}. This means that \sigma_{1} and \sigma_{2} are both permutations of n objects. Since they commute, we can write \sigma_{1} \sigma_{2} = \sigma_{2} \sigma_{1}.

Let us denote the elements in the cycles \sigma_{1} and \sigma_{2} as a_{1}, a_{2}, ..., a_{n} and b_{1}, b_{2}, ..., b_{n} respectively. Then, we can write \sigma_{1} as (a_{1} a_{2} ... a_{n}) and \sigma_{2} as (b_{1} b_{2} ... b_{n}).

Now, applying \sigma_{1} \sigma_{2} gives us (a_{1} a_{2} ... a_{n})(b_{1} b_{2} ... b_{n}) = (a_{1} b_{1}) (a_{2} b_{2}) ... (a_{n} b_{n}). Similarly, applying \sigma_{2} \sigma_{1} gives us (b_{1} b_{2} ... b_{n})(a_{1} a_{2} ... a_{n}) = (b_{1} a_{1}) (b_{2} a_{2}) ... (b_{n} a_{n}).

Since \sigma_{1} and \sigma_{2} commute, we know that (a_{1} b_{1}) (a_{2} b_{2}) ... (a_{n} b_{n}) = (b_{1} a_{1}) (b_{2} a_{2}) ... (b_{n} a_{n}). This means that each