name for an elementary result in group theory
We let a-1
denote the inverse
of an element
a in a group
, ie the the element such that a-1
*a = a*a-1
= e, where e is the identity element
. Then (using the associative property
of the operation
in a group)
(a*b)*(b-1*a-1) = a*(b*b-1)*a-1 = a*a-1 = e
Thus the inverse of a*b is b-1*a-1.
This result is sometimes called the shoes-and-socks theorem because of the following analogy: if you want to undo the act of putting your socks and then your shoes on, you first have to remove your shoes and then take off your socks.