I'm taking an introductory unit on Group Theory at uni and very much enjoy the geometric intuition and interpretations behind the core ideas about groups, however I cannot wrap my head around Lagrange's Theorem, stated in our notes as
Let G be a finite group, and H a subgroup of G. Then |H| divides |G|.
The problem for me comes from the idea of cosets, and how they work. Our notes state that every element of G is in exactly one of the cosets, but if the cosets are the compositions of every element of H (which are therefore also in G) and an element of G then isn't every element in each coset an element of G? I can't convince myself of why an element of G can't be in multiple cosets. There is a proof provided showing that two cosets only intersect non-trivially if they are equal which I can follow without problem, but when I try to convince myself of why they don't intersect I can't explain it abstractly at all.
Any help would be appreciated.