Question

Algebra

"find the numebr of orbits in {1,2,3,4,5,6,7,8} under the subgroup S_8 generated by (1,3) and (2,4,7)"

What I'm having problem understanding is why is (2,7,4) part of the group generated by <(1,3), (2,4,7)>, and also, why is G_{(1,3)(2,4,7)} = |X| - |(1,3)| - |(2,4,7|?

Answer 1

let G be a finite group acting on a finite set X.

for gG,X^g={xX:x.g=x} and X/G be the set of orbits

and formula for this |X/G|=1/|G|

here we have given that X={1,2,3,4,5,6,7,8} and G=<(13)(247)>

and order of G is =lcm(O(13),O(247))=lcm(2,3)=6

it means in this group we have 6 elements

as the order of group is 6,hence order of the elements of the group may be 1,2,3,6(positive divisors of 6)

element of order 1=I(identity)

elements of order 2=(13)

elements of order 3=(247),(274)

elements of order 6=(13)(247),(13)(274)

now we will X^g of every g belongs to G

|X⁽¹⁾|=8 acc to the definition of X^g

|X⁽¹³⁾|=|(2,4,5,6,7,8)|=6

|X⁽²⁷⁴⁾|=|X⁽²⁴⁷⁾|=|(1,3,5,6,8)|=5

|X^(13)(274)|=|X^(13)(247)|=|(5,6,8)|=3

hence sum of |X^g|=8+6+5+5+3+3=30

so |X/G|=30/6=5

hence we get 5 orbits here acc to the formula given above