Determine whether the following permutations are even or odd. (i) (12)(134)(152)

To determine whether a permutation is even or odd, we need to write it as a product of transpositions and count the number of transpositions. An even permutation is a permutation that can be represented as an even number of transpositions, while an odd permutation can be represented as an odd number of transpositions.

Let's break down the given permutation:

(i) (12)(134)(152)

First, we need to write each cycle as a product of transpositions:

• (12) is already a transposition, so we leave it as is.
• (134) can be written as (13)(14).
• (152) can be written as (15)(12).

So, the permutation (12)(134)(152) can be written as (12)(13)(14)(15)(12).

Now, we count the number of transpositions. We have 5 transpositions, which is an odd number.

Therefore, the permutation (12)(134)(152) is an odd permutation.