Clifland

2021-01-31

In an abstract algebra equation about groups, is "taking the inverse of both sides of an equation" an acceptable operation? I know you can right/left multiply equations by elements of the group, but was wondering if one can just take the inverse of both sides?

svartmaleJ

Skilled2021-02-01Added 92 answers

Recall these facts about the groups

Let A be a set,

- an element b is a left inverse for a if

- an element c is a right inverse for a if

- an element is an inverse(or two-sided inverse) for a if it is both a left and right inverse for a.

So in an abstract algebra equation about groups, taking the inverse of both sides of an equation is a valid statement. Since the inverse of an element exist iff its right inverse and left inverse exist. Consider an example.

Let G be the group with identity element e and binary operation

Let a,b in G consider the equation

Case 1

Case 2:

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