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

Recall these facts about the groups
Let A be a set, $×a$ binary operation on A, and $a\in A$. Suppose that there is an identity element e for the operation. Then
- an element b is a left inverse for a if $b×a=e$,
- an element c is a right inverse for a if $a×c=e$,
- 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 $a×x=b$. To solve the equation, take inverse of a both sides.
Case 1
${a}^{-1}\cdot a\cdot x={a}^{-1}\cdot b$
$⇒e\cdot x={a}^{-1}\cdot b$
$⇒x={a}^{-1}\cdot b$
Case 2:
$a\cdot x\cdot {a}^{-1}=b\cdot {a}^{-1}$
$⇒a\cdot {a}^{-1}\cdot x=b\cdot {a}^{-1}$
$⇒e\cdot x=b\cdot {a}^{-1}$
$⇒x=b\cdot {a}^{-1}$

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