Thread: Abstract Algebra proof help

The problem (this is from Gallian btw) states:

Suppose that H is a subgroup of and that H contains (12) and (234). Prove that

The books gives a "solution" to this problem in the back, but I'm having trouble understanding it.

"By closure, (234)(12)=(1342) belongs to H so that |H| is divisible by 3 and 4 and divides 24."

Can someone explain the bolded part? Then maybe I can figure out the rest of what the book says...

Sure.

Recall that in a finite group , the order of an element divides the order of , i.e. .

The cycle (234) is contained in , and as this cycle has order 3, we have that 3 must divide . Similarly, the cycle (1342) is an element of , and so 4 must divide .

And you also know (from Lagrange's theorem) that the order of a subgroup divides the order of the group, which is why the order of must divide the order of , which is 24.