There is no surjective homomorphism from $S_4$ to $D_4$.

Question

I need to know why a surjective homomorphism $: _4 → _4$ does not exist.

I know that S4 has 24 elements and D4 has 8 elements. Do the elements play a part in finding the answer? Do they need to have the same structure?

Answer 1

Hint: It's sufficient to prove that $S_4$ has no normal subgroup of order $3$.

Answer 2

In a homomorphism, the order of the image of an element divides the order of the element. In $S_4$ there are $8$ elements of order $3$; but $D_4$ hasn't got any element of order $3$, so the $8$ elements must be sent to $1$ by the homomorphism, namely they must lie in the kernel, which therefore has order greater than $8$. By the first homomorphism theorem, the image of $S_4$ has order less than $24/8=3$, so it cannot be the whole $D_4$.