# Tour:Factsheet two (beginners)

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To summarize some of the things we have seen so far:

• The power of the structure of groups stems largely from a combination of the associativity, existence of identity element (neutral element) and the existence of inverses.
• The uniqueness of the identity element does not require the use of associativity. However, all the other good structure of groups depend on associativity: including the uniqueness of inverses, the fact that we can drop parenthesization when writing products, the fact that we can cancel elements, the fact that we can solve equations in terms of elements, the involutive nature of the inverse map, and others.
• We can do many special things with finite groups, by combining the fact that there are finitely many elements, and the ability to cancel.
• For a subset of a finite group to be a subgroup, we only require that it be multiplicatively closed. The statement no longer remains true for infinite groups.
• The left quotient and right quotient expressions can be used to test whether a subset is a subgroup.