The two-line symbols for permutations p and q in S5 are p= ( 1 2 3 4 5 ) and q =

Question

The two-line symbols for permutations p and q in S5 are

p= ( 1 2 3 4 5 )     and q = ( 1 2 3 4 5 )

      5 4 1 2 3                        5 3 2 4 1

Write down the permutations p, q, p ^-1, q^-1 in cycle form. Determine the permutations p o q and q o p in cycle form, and write down the orders of p, q,   p o q , q o p.

Write p and q as composites of transpositions, and state the parity of p and q. Let r be the permutation in S5 whose cycle form is (345). Find, in cycle form, the permutation s = r o p o r^-1, that is , the conjugate of p by r. Determine, in cycle form, all the elements of S5 that conjugate p to s.

Thanks for help !!!!

Explanation / Answer

Hi.....

The permutation p maps (12345) to (54123), from the two-line notation given.

In cycle notation, p can be written as a product of disjoint cycles as

(135)(24)   (NB, start with any element and see where it maps to iteratively until you get back to the starting place - eg here, 1 is mapped to position 3 which is then mapped to position 5 which is then mapped back to position 1; 2 is mapped to position 4 which is then mapped back to position 2. Place brackets around the cyclic groups. All cyclic groups are found when all elements are included).

The permutation q maps (12345) to (53241), from the two-line notation given.

In cycle notation, q can be written as (15)(23)(4).

p^-1 and q^-1 are the inverses of each of the permutations. Since p maps (12345) to (54123), its inverse is given by

p^-1 : (12345) -> (34521) (applying p here would get us back to (12345) )

Similarly, q^-1 : (12345) -> (53241) (that is, q is its own inverse, evident since the cycles are in pairs and singles only)

In cycle form then, p^-1 = (153)(24) (the cycles are reversed in direction) and

q^-1 = q = (15)(23)(4) (again the cycles are reversed in direction, which in this case has no effect since they are bicycles or unicycles).

The o symbol indicates a composite operation, where the righthand-side operation is performed first, so that p o q is (12345) -> (53241) -> (14532) (q is performed first, then p on the result). Similarly, q o p is (12345) -> (54123) -> (31425).

In cycle form then, p o q is (1)(2534) and q o p is (1243)(5).

The order of a permutation is defined as the least common multiple (lcm) of the size of each cyclic group. For permutations p, q, p^-1, q^-1, p o q and q o p respectively then, the orders are 6, 2, 6, 2, 4 and 4.

A transposition is a switch of two elements only. If p : (12345) -> (54123) then as a composite of transpositions p can be expressed as (35)(13)(24). Taking this through by each switch in turn

(12345) -> (12543) since elements 3 and 5 switch -> (52143) since elements 1 and 3 switch -> (54123) since elements 2 and 4 switch. Since p can only be expressed by an odd number of transpositions, the parity of p is odd.

Similarly, q can be expressed as for example the composite set of transpositions

(15)(23) (consider the switching steps (12345) -> (52341) -> (53241) ). Since q can only be expressed by an even number of transpositions, the parity of q is even.

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