TITLE: Can the associative property of a group be followed from its other properties?
QUESTION [7 upvotes]: Assume that we have a group together with inverse elements for all its group members, the closure property and also the identity. Can we follow from these properties that our group also has to fulfill associativity?
EDIT:
Or to reformulate my question: If I can set up an arbitrary multiplication table which is complete and closed, so that in each of its rows and columns every element of the "group" is appearing once and only once, does this define me a group automatically? This is really puzzling me..

REPLY [2 votes]: There are lots of loops which are not groups, for example the multiplicative structure of a non-associative algebra such as the octonions $\mathbb{O}$ or any interesting Lie algebra.