Oct 29, 2018 · Here I give a general counting formula for the number of each cycle type in $S_n$ based on combinatorics.

Nov 20, 2024 · The cycles are always listed in a logical and clear order-descending order of their smallest element. If the permutation fixes an element (i.e., it remains unchanged), it is often …

The possible cycle types of elements in S5 are: identity, 2-cycle, 3-cycle, 4-cycle, 5-cycle, product of two 2-cycles, a product of a 2-cycle with a 3-cycle. These have respective orders 1; 2; 3; 4; …

A problem from my algebra homework requests the following: List all the possible cycle structures in $S_6$. For each cycle structure, compute the order of an element with that cycle structure.

Nov 19, 2015 · For your example, in $S_8$ you want a product of two 4-cycles: $ (a_1 \, a_2 \, a_3 \, a_4) (a_5 \, a_6 \, a_7 \, a_8)$. Start by counting the number of ways to set up the …

There are two possible cycle structures for an element of order $3$ in $S_ {6}$, as you noted yourself. Hence we need to count the conjugates of $x = (123)$ and of $y = (123) (456)$ in …

Jan 7, 2016 · Elements of order two can either occur as $2$-cycles or as products of $2$-cycles of the form $ (ab) (cd)$. The number of unique $2$-cycles is $6 \cdot 5 / 2 = 15$.

Feb 17, 2012 · How many elements of order 6 are in $S_5$? Here is my solution: possible cycle structures are $ [2,3]$. Applying class equation gets $5! \over 2!3! $ = $10$. Is my solution …

Each chain closes upon itself, splitting the permutation into cycles. The cycles can be written in any order. Within each cycle, we can start at any number. There are two conventions for …

The remaining elements are either products of two disjoint transpositions, such as (12)(34), in which case we’re done, or four cycles such as (1234). The latter can be written (12)(23)(34).