I need help in the mapping problems..
Please help!
Q1) A) Write (a1 a2 a3.....ak) ^ (-1) in cyclic notation (without the symbol
for inverse)?
B) For which values of k will every k-cycle be its own inverse?
Here: ^ (-1) means inverse of the set.
Thanks in advance!
This is what I got...is this fine??
The inverse for the k-cycle
( a[1] a[2] ... a[k] )
is
( a[k] a[k-1] a[k-2] ... a[2] a[1] )
= ( a[1] a[k] a[k-1] a[k-2] ... a[2] )
Therefore a k-cycle is its own inverse if and only if
- If there is no a[2], ( a[1] ) = identity permutation is obviously its own inverse. k = 1.
- Otherwise, there is an a[2], and we must have a[2] = a[k] → k = 2 since a number may only appear once in a cycle.
So, k = 1 or 2.