1. The problem statement, all variables and given/known data Not a homework question actually. I'm having trouble understanding some of the corollaries to Lagrange's theorem. Theorem: Let H b e a subgroup of a finite group G. then |H| divides |G| Corollary 1: if g is an element of a finite group G, then |g| divides |G|. proof: the cyclic subgroup |H| = <g> generated by g has |H| = |g|. question: how do we know such a cyclic subgroup H exists as required by the proof? Corollary 2: If p is a prime, then every group G of order p is prime. proof: write H = <g>. Then |H| divides |G| so |H| is 1 or |H| = p = |G|. question: Again, how do we know that such a cyclic subgroup H exists in the first place?