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The given question involves a permutation \(\alpha = (14)(23)(34)(24)(12)\), and asks about the parity (odd/even) of \(\alpha^{1000}\) and \(\alpha^{1001}\).

To solve this, we need to follow these steps:

1. **Simplify the permutation \(\alpha\)**: 
   The permutation is written as a product of disjoint cycles: 
   \[
   \alpha = (14)(23)(34)(24)(12)
   \]
   We simplify this by following the effects of each transposition.

   Let's break down each transposition:
   - Start with \( (14) \), which swaps 1 and 4.
   - Then \( (23) \), which swaps 2 and 3.
   - \( (34) \) swaps 3 and 4.
   - \( (24) \) swaps 2 and 4.
   - Finally, \( (12) \) swaps 1 and 2.

   After simplifying all these transpositions, we can see that the permutation \(\alpha\) becomes the identity permutation. That is, \(\alpha\) leaves every element unchanged.

2. **Check powers of \(\alpha\)**:
   Since \(\alpha\) is the identity permutation, its powers \(\alpha^{1000}\) and \(\alpha^{1001}\) will also be the identity permutation. The identity permutation is **always an even permutation** because it can be written as a product of zero transpositions, and zero is an even number.

Thus:
- \(\alpha^{1000}\) is an even permutation.
- \(\alpha^{1001}\) is also an even permutation.

The correct answer is **Option 1**: \(\alpha^{1000}\) is an even permutation and \(\alpha^{1001}\) is an even permutation.

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Would you like more details on the simplification of the transpositions or the concept of odd and even permutations?

Here are some related questions:
1. What is the formal definition of an odd and even permutation?
2. How can you determine if a permutation is even or odd by the number of transpositions?
3. What is the role of disjoint cycle decomposition in determining the parity of a permutation?
4. How are powers of permutations generally calculated?
5. Can any permutation be decomposed into a product of transpositions?

**Tip**: When working with permutations, it is helpful to simplify them by applying the transpositions step by step to see their overall effect on elements.

Let α = (14)(23)(34)(24)(12). Which of the following statements is true regarding α^1000 and α^1001?

Explore the parity of permutation α = (14)(23)(34)(24)(12) and analyze its powers α^1000 and α^1001. This problem covers permutation parity (odd/even) and the role of the identity permutation. Suitable for university-level abstract algebra students.

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