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Maths Lover
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first ,
if p is prime , show that an element has order p in Sn iff it's cycle decomosition is a product of commuting p-cycles
my solution is very diffrent about the one in the book and I don't know if my strategy is right
my proof
______
let T is an element of Sn
and the cycle decomposition of T = t1 t2 t3 ... tm
t1 , t2 , ... , tm are disjoint cycle
so
ti tj = tj ti for all i,j
T^p = ( t1 t2 ... tm )^p = (t1)^p (t2)^p ... (tm)^p
now , ti , tj are 2 disjoint cycle so if ti or tj are not equal to identity permutation then ,
ti tj =\= e
now if T^p = 1
then
(t1)^p (t2)^p ... (tm)^p = 1
but (t1)^p , (t2)^p , ... , (tm)^p are disjoint
so if (ti)^p =\= 1 for all i then
(t1)^p (t2)^p ... (tm)^p =\= 1
but (t1)^p (t2)^p ... (tm)^p = 1
so it's nessary that
(ti)^p = 1 for all i
so
l ti l = p
and ti 's are commuting because they are disjoint
so we proved that order of T = p where p is prime then it's cycle decomposition has the same order p
now
every t1 is a p-cycle
is this right ?
the otherway is easy to prove
please help
in this question we can't use the rule which says @ order of an element in Sn is the LCD of the lenghts of the cycles in its cycle decomposition because this rule didn't come in the book and I search for a proof without it
if p is prime , show that an element has order p in Sn iff it's cycle decomosition is a product of commuting p-cycles
my solution is very diffrent about the one in the book and I don't know if my strategy is right
my proof
______
let T is an element of Sn
and the cycle decomposition of T = t1 t2 t3 ... tm
t1 , t2 , ... , tm are disjoint cycle
so
ti tj = tj ti for all i,j
T^p = ( t1 t2 ... tm )^p = (t1)^p (t2)^p ... (tm)^p
now , ti , tj are 2 disjoint cycle so if ti or tj are not equal to identity permutation then ,
ti tj =\= e
now if T^p = 1
then
(t1)^p (t2)^p ... (tm)^p = 1
but (t1)^p , (t2)^p , ... , (tm)^p are disjoint
so if (ti)^p =\= 1 for all i then
(t1)^p (t2)^p ... (tm)^p =\= 1
but (t1)^p (t2)^p ... (tm)^p = 1
so it's nessary that
(ti)^p = 1 for all i
so
l ti l = p
and ti 's are commuting because they are disjoint
so we proved that order of T = p where p is prime then it's cycle decomposition has the same order p
now
every t1 is a p-cycle
is this right ?
the otherway is easy to prove
please help
in this question we can't use the rule which says @ order of an element in Sn is the LCD of the lenghts of the cycles in its cycle decomposition because this rule didn't come in the book and I search for a proof without it