In this question on yahoo answers , the answer says ,
"with $t = 6$, then there are 6 * (5 - 1) = 24 elements of order $5$ "
my question is , how did " 6 * ( 5 - 1 ) " come from ?
Which theorem is used here ?
The same question for " with $s = 10$, there are at least 10 * (3-1) = 20 elements of order $3$"
In dummit and foote , 3rd ed , in page $145$
in prop $21$ which proves that if $G$ is a group of order 60 and $G$ has more than one sylow 5-subgroup then G is simple
The proof says ,
suppose the oposite ,
let $H$ be a proper normal subgroup of $G$ if $5$ divides $ |H| $ then $H$ contains a sylow 5-subgroup of $G$ and since $H$ is normal then it contains all the conjugates of this sylow 5-subgroup , so , in particular $ |H|$ is bigger than or equal to $ ( 1+(6 . 4) ) = 25$
My question is , how did the calculation " $ |H|$ is bigger than or equal to $ ( 1+(6 . 4) ) = 25$ " come from ?