1. ## Math Help

Need help proving that even permutations form a group?

where the say of even permutations in symmetric group of degree n(S sub n) forms a subgroup of S sub n.

2. Originally Posted by biggybarks
Need help proving that even permutations form a group?

where the say of even permutations in symmetric group of degree n(S sub n) forms a subgroup of S sub n.
.

3. Originally Posted by biggybarks
Need help proving that even permutations form a group?

where the say of even permutations in symmetric group of degree n(S sub n) forms a subgroup of S sub n.

What have you done, where are you stuck? The group of even permutations in n objects is denoted $\displaystyle A_n$ and called the alternating group.

Tonio

4. ## math help

having a hard time getting started on it. Don't know where to begin

5. Here is a rough outline of the proof. Fill in the blanks.

1. Suppose that $\displaystyle \sigma,\tau\in A_n$ then $\displaystyle \sigma,\tau$ are both the product of an even number of transpositions. So argue that $\displaystyle \sigma\tau$ is also the product of an even number of transpositions.

2. Let $\displaystyle \sigma\in S_n$ be a transposition. What is $\displaystyle \sigma^2$? What does that imply?

3. Let $\displaystyle \tau\in S_n$ such that $\displaystyle \tau=\prod_{i=1}^n t_i$ where $\displaystyle t_i$ are transpositions. Using 2. what is $\displaystyle \tau^{-1}$ in terms of $\displaystyle \prod_{i=1}^n t_i$? (if this doesn't make sense...try doing some small example and see the pattern)

6. ## help

still lost. Don't really understand how to get this done

7. Originally Posted by Drexel28
Here is a rough outline of the proof. Fill in the blanks.

1. Suppose that $\displaystyle \sigma,\tau\in A_n$ then $\displaystyle \sigma,\tau$ are both the product of an even number of transpositions. So argue that $\displaystyle \sigma\tau$ is also the product of an even number of transpositions.

2. Let $\displaystyle \sigma\in S_n$ be a transposition. What is $\displaystyle \sigma^2$? What does that imply?

3. Let $\displaystyle \tau\in S_n$ such that $\displaystyle \tau=\prod_{i=1}^n t_i$ where $\displaystyle t_i$ are transpositions. Using 2. what is $\displaystyle \tau^{-1}$ in terms of $\displaystyle \prod_{i=1}^n t_i$? (if this doesn't make sense...try doing some small example and see the pattern)
Originally Posted by biggybarks
still lost. Don't really understand how to get this done
I think you should be able to handle one. For the second one merely note that a transpotition $\displaystyle \sigma=(ab)$ is its own inverse. Therefore $\displaystyle \sigma_e=\sigma^2=(ab)(ab)$ so that $\displaystyle A_n$ contains the identity.

If $\displaystyle \tau=t_1\cdots t_n$ are are transpositions then $\displaystyle \tau^{-1}=t_n\cdots t_1$. To see this merely note that $\displaystyle t_1\cdots t_n\cdot t_n\cdots t_1=t_1\cdots t_{n-1}\cdot t_{n-1}\cdots t_1=\cdots=t_1\cdot t_1=e$. Therefore an even permutation has...........what?