Mark each of the following true or false. a. Every permutation is a cycle. b. Every cycle is a permutation.
The correct answer and explanation is:
Sure! Here are the answers with explanations:
a. Every permutation is a cycle.
False
b. Every cycle is a permutation.
True
Explanation:
A permutation of a set is any rearrangement of its elements. For example, if you have the set {1,2,3}\{1, 2, 3\}, then the permutation that sends 1→2, 2→3, and 3→1 is a rearrangement of these elements.
A cycle is a specific type of permutation where elements are rotated in a closed loop. For instance, the cycle (1 2 3)(1\,2\,3) means 1 goes to 2, 2 goes to 3, and 3 goes back to 1.
- Why every cycle is a permutation:
Since a cycle just rearranges elements by cycling them, it is indeed a permutation. Cycles are a special case of permutations where only some elements are moved, and the rest remain fixed (or the whole set is rotated in a loop). Thus, all cycles are permutations. - Why not every permutation is a cycle:
Permutations can be more complicated than single cycles. Any permutation can be decomposed into one or more disjoint cycles. For example, the permutation on {1,2,3,4}\{1, 2, 3, 4\} that sends 1→2, 2→1, 3→4, and 4→3 is composed of two cycles: (1 2)(1\,2) and (3 4)(3\,4). This permutation is not a single cycle but a product of two disjoint cycles.
In summary:
- Every permutation can be expressed as a product of cycles, but it is not necessarily just one cycle.
- Every cycle rearranges elements, so it is a permutation.
Additional note:
In group theory, the set of all permutations of nn elements forms the symmetric group SnS_n. The cycles are the building blocks for elements of this group. Understanding this distinction helps in studying the structure and properties of permutations.