# Abstract Algebra/Group Theory/Group/Definition of a Group/Definition of Inverse

Inverse:
1. if c is in G, c-1 is in G.
2. c*c-1 = c-1*c = eG

# Definition of Inverse

Let G be a group with operation ${\displaystyle \ast }$

${\displaystyle \forall \;g\in G:\exists \;g^{-1}\in G:g\ast g^{-1}=g^{-1}\ast g=e_{G}}$

# Usages

1. If g is in G, g has an inverse g−1 in G
2. b is the inverse of g on group G if
b is in G, and
b ${\displaystyle \ast }$ g = g ${\displaystyle \ast }$ b = eG.
eG here again means the Identity of group G.
3. If b is the inverse of g on group G, then
b is in G, and
b ${\displaystyle \ast }$ g = g ${\displaystyle \ast }$ b = eG.

# Notice

1. G has to be a group