# 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.

- The
G is closed under *.
- The
* is associative.
- 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.
- 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