Question

abstract algebra: what is the difference between an "orbit" and a "cycle" regarding permutations?

Answer 1

cycle goes on forever, like trig functions. One cycle is from 0 to 2pi, but orbit continuously goes around same path?

Answer 2

is an orbit a collection of cycles that define a single permuation? all orbits are cycles.

a cycle is a set of directions, and can suggest more than a one step permutation?

Answer 3

\[\sigma=\begin{pmatrix}1&2&3&4&5\\3&5&4&1&2\end{pmatrix}=(134)(25)\]

Answer 4

the orbits are: (134) and (25); or is that {1,3,4} and {2,5} proper notation?

Answer 5

product of 3 cycles
\[\mu=(126)(24)(31)=\begin{pmatrix}
1&2&3&4&5&6\\
3&2&1&4&5&6\\
3&4&1&2&5&6\\
3&4&2&6&5&1\\
\end{pmatrix}=
\begin{pmatrix}
1&2&3&4&5&6\\
3&4&2&6&5&1\\
\end{pmatrix}\]

Answer 6

i think the difference is that orbits are sets
the ideas are the same

Answer 7

also an "orbit" can be on element, whereas i don't think you count (a) as a cycle

Answer 8

*one element

Answer 9

is the proper notation for an orbit a roster set notation then? {a,b,c,...,n}

Answer 10

also, product of cycles, if seen them go left to right, and other right to left ... not sure if there is a proper direction for those