# On the properties of non-commutative groups

## Homework Statement

Let $$[G,+,0]$$ be a non-abelian group with a binary operation $$+$$ and a zero element $$0$$.

To prove that if both the zero element and the inverse element act on the same side, then they both act the other way around, that is:
If $$\forall a \in G$$,
$$a + 0 = a$$,
and
$$a + (-a) = 0$$,
then it can be proven that
$$0 + a = a$$
and
$$(-a)+ a = 0$$.

## Homework Equations

This is not really homework, it is just something that has been bothering me. The doubt in question arises because certain books (like Herstein's Modern Algebra) define groups as structures in which both the inverse and the zero element act on both sides, regardless of if the group is commutative or not. On this subject my linear algebra professor said that having the zero and the inverse act on the same side is equivalent to having them act on both, since the latter can be proven from the former, hence my question.

## The Attempt at a Solution

I have been trying to do what is stated in section 1, but I only end up concluding tautologies like 0=0 or so.

Thanks in advance for your help.
(By the way, I can't get the latex command support thing we have in here do display { and })

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
Latex uses { and } internally to group symbols. To get them to display, use \{ and \}.

@SrEstroncio
The existence of inverses, properties of the inverse operation, and the two equations you listed are sufficient to prove what you wanted to prove.
Remember: $$-(a+b)=-b+(-a)$$ $$\forall a,b \in G$$