Tour:Factsheet two (beginners)

We're working on a survey article called manipulating equations involving groups, which gives the leading ideas for how to manipulate equations involving groups, how this differs from manipulating ordinary equations, and how the group being Abelian affects the manipulation. In the meantime, here's an easy factsheet:

• 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: including the uniqueness of inverses, and the fact that we can cancel elements, stems from a combination of associativity and the existence of identity elements.
• 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.