# Thread: Abstract Algebra proof help

1. ## Abstract Algebra proof help

The problem (this is from Gallian btw) states:

Suppose that H is a subgroup of $\displaystyle S_4$ and that H contains (12) and (234). Prove that $\displaystyle H=S_4$

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...

2. Sure.

Recall that in a finite group $\displaystyle H$, the order of an element $\displaystyle g\in H$ divides the order of $\displaystyle H$, i.e. $\displaystyle \text{ord}(g)\ |\ |H|$.

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

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 $\displaystyle H$ must divide the order of $\displaystyle S_4$, which is 24.