Given a group G and its subgroup H that creates cosets, prove that cosets form a group iff H - is a normal subgroup. I've tried to find any good prove but I failed - most of the sources (including ProveWiki) provide this statement "as is" and other sources provide incorrect proves like here:
https://www.youtube.com/watch?v=vYKdh5oQ4Zw&list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6&index=9&t=502s
6:54 - brackets are opened as the operation is commutative - but in fact, it not necessarily true.
Can you, please, tell why there's a huge leap between statement that
product of two elements from two cosets must be contained by the third coset and this operation should be associative <=> subgroup ability to form up equal left and right cosets.
Or give me, please, prove of the statement, I'll try to understand the leap.
