# Group theory: am I allowed to “multiply” both sides of an equation with the same element?

First of all, I'm not a mathematican in any sense, I'm just curiuous sometimes. So please forgive me for not being rigorous in stating my actual problem.

Given a (non-abelian) group with its usual axioms, consisting of the set $$G$$ (lets call the group also $$G$$) and an operation $$\cdot$$, do I have to proof that from $$h\cdot g=e$$ follows that $$(h\cdot g)\cdot g^{-1}=e\cdot g^{-1}$$ , where $$h,g \in G$$ and $$g^{-1}$$ is the inverse element of $$g$$ and $$e$$ is the neutral element?

In principle this question is very similar to

Algebra: What allows us to do the same thing to both sides of an equation?

but if I got it right, a group doesn't need to be composed of "numbers". It can be anything, such as a rotation maybe or a far more abstract object I can't even imagine

So basically my question is, if the answer to the related question applies to any group in general or if I have to prove this somehow?

Thanks in advance and sorry for my poor english- and math-skills.

• Yes, it works also for "non numbers" (as per many similar question linked to the linked question); it is enough that the operation is defined between the objects you are dealing with, and it is always true that : "if $a=b$, then $(a \text { op } c = b \text { op } c)$. See rules for equality. – Mauro ALLEGRANZA Oct 21 '19 at 15:12
• Yes! The inverse in a group is unique! Assume $x^{-1}=a^{-1}$. The definition of the inverse gives us $x^{-1} x = e$ and if $x^{-1}=a^{-1}$ that is equivalent to $xa^{-1} =e$ and multiplying with $a$ gives $x=a$. – cptflint Oct 21 '19 at 15:19
• Doesn't even need to be a group. Applies to everything. If $a$ and $b$ are two names for the same thing, and "$manipulate()$" is a manipulation that has a consistent result determined by its input then $manipulate(a)=manipulate(b)$... Admittedly abstract mathematics can into great detail as to how we can define,interpret, and prove such simple concepts of say "a thing is itself" but that's fairly advanced and abstract. – fleablood Oct 21 '19 at 15:25
• @cptflint although that is what the op was eventually trying to show, his/her question was much more basic. S/he was asking how we can multiply both sides by $a$ and if that is something that needs to be prove. If $M=N$ then do we know $Ma=Na$. – fleablood Oct 21 '19 at 15:28
• Subtle side issue: To say: if $a=b$ then $manipulate(a)=manipulate(b)$ requires that "$manipulate()$" is well-defined and consistent. I think some confusion comes from things such as. If $a=b$ then $a^2=b^2$ and $a*0 =b^2$ is true. But if $a^2=b^2$ or $a*0=b*0$ then we can't say $a=b$. But this is because $manipulate$ means "square it" or "multiply" by zero, is well defined and consistent. The concept of "find a number that when squared is equal to the input" or "find a number that when multiplied by zero is the input" are NOT well-defined and are inconsistent. – fleablood Oct 21 '19 at 15:35

If we have, say, $$a=b$$ then for any function $$f$$, any function at all, we have $$f(a)=f(b)$$ For two values that are the same, inputting them into $$f$$ will give you the same output, because $$f$$, being a function, evaluates to a unique value on a given input. Multiplying by $$g^{-1}$$ on the right is a function. Calling this function $$m_g$$, we have $$m_g(x) = x\cdot g^{-1}$$ For your equation, we are saying that $$h\cdot g = e$$ so $$(h\cdot g)\cdot g^{-1} = m_g(h\cdot g) = m_g(e) = e\cdot g^{-1}$$ Therefore the manipulation is valid.