The permutations P = p and q in So have the two-line symbols 1 2 3 4 5 6 15 2 3 4 6 1 2 3 4 5 6 4 6 2 1 5 3, and q = (a) Write down the permutations p, q, p¯¹ and q¯¹ in cycle form. (b) Write down the permutations poq and qop in cycle form. (c) Write down the orders of p, q, po q and q o p. (d) Write down p and q as composites of transpositions, and state the parity of p and the parity of q. (e) Let r be the permutation in S6 whose cycle form is (4 6 5). Write down. in cycle form, the permutation t=ropor-¹ (the conjugate of p by r). (f) Write down, in cycle form, the elements of the cyclic subgroup H of S6 generated by p. (g) Write down, in cycle form, the elements of the conjugate subgroup
The permutations P = p and q in So have the two-line symbols 1 2 3 4 5 6 15 2 3 4 6 1 2 3 4 5 6 4 6 2 1 5 3, and q = (a) Write down the permutations p, q, p¯¹ and q¯¹ in cycle form. (b) Write down the permutations poq and qop in cycle form. (c) Write down the orders of p, q, po q and q o p. (d) Write down p and q as composites of transpositions, and state the parity of p and the parity of q. (e) Let r be the permutation in S6 whose cycle form is (4 6 5). Write down. in cycle form, the permutation t=ropor-¹ (the conjugate of p by r). (f) Write down, in cycle form, the elements of the cyclic subgroup H of S6 generated by p. (g) Write down, in cycle form, the elements of the conjugate subgroup
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A,b,c
Transcribed Image Text:The permutations
p and q in So have the two-line symbols
1 2 3 4 5 6
4 6 2 15 3,
(3)
1 2 3 4 5 6
1 5 2 3 4 6
P =
and q =
(a) Write down the permutations p, q, p
and q¹ in cycle form.
(b) Write down the permutations po q and q o p in cycle form.
(c) Write down the orders of p, q, po q and qop.
(d) Write down p and q as composites of transpositions, and state the
parity of p and the parity of q.
(e) Let r be the permutation in S6 whose cycle form is (4 6 5). Write down,
in cycle form, the permutation t=ropor-¹ (the conjugate of p by r).
(f) Write down, in cycle form, the elements of the cyclic subgroup H of S6
generated by p.
(g) Write down, in cycle form, the elements of the conjugate subgroup
roHor-¹.
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