I have just started studying group theory, and we have naturally looked at Lagrange's theorem and stated Cauchy's as a collorary.
Lagrange: If $ H \subset G$ we must have $ |G| = |G:H| |H| $. Or shortly, the order of every subgroup must divide the order of the parent group. However, as far as I understand this doesn't say that if |G| = 0 (mod n), there must be subgroups of order n.
Cauchy: If the order of a group factorises $ |H| = p_1*p_2*p_3...$ there must be elements of order $p_1,p_2...$.
How comes one of them gives a mere can while the other one gives a must? I have tried looking up separate proofs to Cauchy, but they seem to surpass my group theory knowledge. In lectures we proved Lagrange by proving that every element is in one, and only one, coset. How can we easily prove Cauchy's theorem?
The Cauchy Theorem is only true for primes. In fact the group $\mathbb{Z}_2 \times \mathbb{Z}_2$ has order $4$, but no element of order $4$.
The Lagrange Theorem tells you that a subgroup of certain order exists, then it's order must divide the order of the group. The Cauchy Theorem is some sort of a reverse implication of the Lagrange Thereom, as it works only for prime divisors of the order of the group.