Given $G = A_6$, where $A_6$ is the alternating group on 6 elements, we need to determine the six possible cycle structures of the elements in $A_6$. The alternating group $A_6$ consists of all even permutations of the symmetric group $S_6$, meaning only permutations that can be expressed as the product of an even number of transpositions.

Step-by-Step Process to Determine the Cycle Structures

1. Permutation Types in $S_6$: In $S_6$, the elements are made up of various types of cycles. We classify these based on their cycle lengths:

   • 1-cycle: This represents the identity element.
   • 2-cycle: A transposition (not in $A_6$ because it is odd).
   • 3-cycle: A permutation involving three elements.
   • 4-cycle: A permutation of four elements.
   • 5-cycle: A permutation of five elements.
   • 6-cycle: A permutation of six elements.
   • Products of cycles: Some permutations can be written as products of disjoint cycles (e.g., a 2-cycle and a 3-cycle).

2. Allowed Even Permutations in $A_6$: Since $A_6$ contains only even permutations, odd-length cycles alone (such as 1-cycles and transpositions) are excluded unless they are combined with others to make an even permutation.

The Six Possible Cycle Structures in $A_6$:

1. Identity element (denoted as $(1)$): The trivial cycle where no elements are permuted. This is always present in any group.

2. Three 2-cycles (denoted as $(a\ b)(c\ d)(e\ f)$): A product of three disjoint 2-cycles. This is an even permutation because the number of transpositions is even.

3. 3-cycle (denoted as $(a\ b\ c)$): A single 3-cycle that permutes three elements, which is an even permutation.

4. Product of a 3-cycle and a 2-cycle (denoted as $(a\ b\ c)(d\ e)$): This is the product of a 3-cycle and a disjoint 2-cycle, which is an even permutation.

5. 5-cycle (denoted as $(a\ b\ c\ d\ e)$): A single 5-cycle that permutes five elements. This is an even permutation.

6. Double 3-cycle (denoted as $(a\ b\ c)(d\ e\ f)$): A product of two disjoint 3-cycles, which is an even permutation since it involves two transpositions.

Summary of the Six Cycle Structures in $A_6$:

1. Identity element: $(1)$
2. Product of three 2-cycles: $(a\ b)(c\ d)(e\ f)$
3. Single 3-cycle: $(a\ b\ c)$
4. Product of a 3-cycle and a 2-cycle: $(a\ b\ c)(d\ e)$
5. Single 5-cycle: $(a\ b\ c\ d\ e)$
6. Product of two 3-cycles: $(a\ b\ c)(d\ e\ f)$

These are the six possible cycle structures of the elements in $A_6$.

---

Would you like more details or further explanations on any of the steps?

Here are five related questions for further exploration:

1. What are the orders of each element type in $A_6$?
2. How do the cycle structures of elements in $A_6$ differ from those in $S_6$?
3. What is the order of the alternating group $A_6$?
4. Can you describe the conjugacy classes of $A_6$?
5. What is the significance of $A_6$ being a simple group?

Tip: Understanding cycle notation and transpositions is essential for working with permutations in symmetric and alternating groups!