Question

OTO (7) (a) Let T = (a1, ..., ak) be a k-cycle in Sn, and let o E Sn. Prove that is the k-cycle (o(a), o(az),..., 0(ak)) (b) Let o,t e Sn. Prove that if t is a product of r pairwise disjoint cycles of lengths k1,..., kr, respectively, where kit..., +kr = n, then oto-1 is also a product of r pairwise disjoint cycles of lengths k1,..., kr. (c) Let T1 and T2 be permutations in Sn. Prove that T1 and T2 are conju- gate if and only if, for some positive integer r, there is a partition of n into r parts, n = ki+...+kr, such that both Ti and T2 can be written as the product of r pairwise disjoint cycles of lengths k1, ..., kr.

Answer 1

(a) in this part it's enough to show mapping of $\sigma \tau \sigma ^-^1$ on set [n] is same as given cycle.

. to T= (9,92, •96) 9 k-cycle in Sn. Firstly note that for for any te[n] \ {9,92. -- ak} I) I(t)= t: ie. I fixes elements other than aj's. (a) To prove that orot- (5(9,), 0(42), -..01ak)). prove this we need to show that 7 for 8(9) € [n] reo maps olain (ait) & for #otto 27 for t&{0191), ...019k)} scot fixes t. these two statements will prove reo is preciesly that (what we're doing here is checking rest on each element of [n]). Proof. » otot (ola;)) = secolocail) but örlai)=ai = o(z) (a;) but (a;)=941. = 6 caiti). so, oro maps. reais to 8(9;+1). (as, in cycle.... for 27. tt [n]) {019;) }; taking to r(t) € [n]) {9,92, ... 0k}. using now consider ozol(t) ollOc)). from I olot(t)) = t. given gives ot so fort so, sro fixes every other element then (619;}) => Torota (019), 6(92), ocak)

(b) it's application of part (a)

(b) te (41, , given, t is product of a paiswise disjoint cycles. of lengths kluka, Let's assume tl.. + ak) (47,92 ,.-. qžs). cãijaa...ar). of length K2 first cycle of length ki where; indeed & each of them is distanct. gith cycle "cycle of length ky i each a; € [n] now; o cota od ), Dz ( 1st cycle and cycle ith cycle. For A since, or=id. put the in bet cycles. 1o ( stycle) or 5 (2 dayle) o to 3rd lyde gol )... so (oth unclejo serta using result (a), (ocail ola,) ... o(ak)). Cola") ola).-61937) rth cycle. ist cycle since, each a ) olary) rró => rest is also a a is distinct 7 8(a), is also distinct (o(a) och olde). - 864'ka) (olan olay length K length kr product of r disjoint by cles of length ka, ko Hence, proved.

(c) If $\tau _1$ and $\tau _2$ are conjugate i.e. $\tau _1=\sigma \tau _2\sigma ^-^1$ for some $\sigma$ belonging to $S_n$ .

in above two parts (a) (b) we showed that $\tau _1$ and $\tau _2$ have same cycle type. In (c) exctly same is stated.