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 Hso 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...