What is a group?

analogy

A group is like a of dance moves.

plain english

Groups are symmetrical structures. Group theory teaches us how to create and understand symmetrical structures.

technical

A group (G,*) is a G combined with a * such that the following axioms hold.

1. The G is closed under *.
2. The * is associative.
3. There is an element of the set G which is called e. For every element g of G, e * g = g * e = g.
   e is the identity of G.
4. For every element g of G, there is an element g' in G such that g * g' = g' * g = e.
   g' (which can also be written as g^-1) is the inverse of g.

example

dance moves

combine moving left and right by doing them one after the other

identity

staying still

inverse

combining the group elements of stepping left and stepping right leads to the identity, staying still