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Let $G$ be a nonempty set closed under an associative product, which in addition satisfies :

A. There exists an $e$ in $G$ such that $a \cdot e=a$ for all $a \in G$.

B. Given $a \in G$, there exists an element $y(a) \in G$ such that $a \cdot y(a) =e$.

Prove that $G$ must be a group under this product.

Attempt -Since Associativity is given and Closure also, also the right identity and right inverse is given .So i just have to prove left identity and left inverse.

Now as $ae=a$ post multiplying by a, $aea=aa$. Now pre multiply by a^{-1} I get hence $ea=a$. And doing same process for inverse Is this Right?

Martin Sleziak
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Taylor Ted
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2 Answers2

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Let, $ab=e\land bc=e\tag {1}$ for some $b,c\in G$. And, $ae=a\tag{2}$ From $(2)$, $$eae=ea\implies(ab)a(bc)=ea\implies ((ab)(ab))c=ea\implies ec=ea\tag{3}$$

Similarly, $$ae=a\implies a(bc)=a\implies (ab)c=a\implies ec=a\tag{4}$$

$(3)$ and $(4)$ implies, $$ea=a$$

Also from $(3)$ and $(1)$, $$(bab)(bca)=e\implies b((ab)(bc)a)=e\implies ba=e$$

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    (1) is wrong, I think, since you pre-suppose that actually there is a left inverse. Where did you find that? It might be the case that no left inverse exists for any element. How do you prove existence? –  Jul 27 '17 at 08:08
  • Note that given $a\in G$ there exists an element $y(a)\in G$ such that $a\cdot y(a)=e$. Similar is the argument for $b$. In my answer above $y(a)=b$ and $y(b)=c$. Does it help @Jason? –  Jul 27 '17 at 13:31
  • Can you please clarify the last assert $(bab)(bca)=e$? I fail to see how it follows from $(1)$, Thank you! – galra Mar 03 '18 at 01:10
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    @galra: See the edit. Observe that by $(3)$ we have, \begin{align}(bab)(bca)&=(be)(ea)\&=b(ec)&\text{by (3)}\&=(be)c\&=bc\&=e\\end{align}And by $(1)$ we have, \begin{align}(bab)(bca)&=b(ab)(bc)a\&=b(e)(e)a\&=ba\end{align} Hope it helps. –  Mar 03 '18 at 04:35
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1.

$(y(a)\cdot a)\cdot (y(a)\cdot a) = y(a) \cdot (a \cdot y(a))\cdot a = y(a) \cdot e \cdot a=(y(a)\cdot e) \cdot a = y(a) \cdot a$.

$(y(a)\cdot a)\cdot ((y(a)\cdot a) \cdot y(y(a) \cdot a)) = (y(a) \cdot a) \cdot y(y(a) \cdot a)$.

$y(a)\cdot a = e$

2.

$e\cdot a = (a \cdot y(a))\cdot a=a\cdot(y(a)\cdot a)=a\cdot e=a$.

tchappy ha
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