Prove that if the cancellation law holds from both sides in a set, it forms a group.

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November 22nd 2011, 08:31 AM
Nikita2011

Prove that if the cancellation law holds from both sides in a set, it forms a group.

Let G be a finite nonempty set with an operation * such that:
1. G is closed under *.
2. * is associative.
3. Given a,b,c in G with a*b=a*c, then b=c.
4. Given a,b,c in G with b*a=c*a, then b=c.

Prove that G must be a group under *.
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It's obvious that identity element satisfies the conditions 3 and 4, but I don't know whether that proves that the identity element is contained in G or not? moreover, How can I show that the inverse of any element in G is contained in G?
November 22nd 2011, 09:32 AM
FernandoRevilla

Re: Prove that if the cancellation law holds from both sides in a set, it forms a gro

Quote:

Originally Posted by Nikita2011

Let G be a finite nonempty set with an operation * such that: 1. G is closed under *. 2. * is associative. 3. Given a,b,c in G with a*b=a*c, then b=c. 4. Given a,b,c in G with b*a=c*a, then b=c. Prove that G must be a group under *.

Let $a,b\in G$ . Suppose $G=\{a_1,\ldots,a_n\}$ where $a_1,\ldots,a_n$ are distinct. Use the left cancellation law to prove that $G=\{aa_1,\ldots,aa_n\}$ . This implies that the equation $ax=b$ is solvable . Similarly we can prove that the equation $ya=b$ is solvable . Could you continue ?
November 26th 2011, 05:35 AM
Nikita2011

Re: Prove that if the cancellation law holds from both sides in a set, it forms a gro

Quote:

Originally Posted by FernandoRevilla

Let $a,b\in G$ . Suppose $G=\{a_1,\ldots,a_n\}$ where $a_1,\ldots,a_n$ are distinct. Use the left cancellation law to prove that $G=\{aa_1,\ldots,aa_n\}$

Would you explain how you obtained this set?

Quote:

This implies that the equation $ax=b$ is solvable . Similarly we can prove that the equation $ya=b$ is solvable . Could you continue ?

This is how I understood your post: since for any a,b in G $ax=b$ is solvable, then by definition there exists an x that satisfies that relation, so if let a=b, that proves the existence of $e_R$ , the same logic can be deployed to prove the existence of $e_L$ , now it'll be easy to prove that these two are equal. after proving that the left and right identity elements are the same and we denote the identity element by e, now if we let $ax=e$ , since it's solvable, it proves the existence of $a^{-1}_R$ , the same logic can be applied to the equation $ya=e$ and that proves the existence of $a^{-1}_L$ , now again it's easy to verify that these two will be the same element.
November 26th 2011, 06:06 AM
emakarov

Re: Prove that if the cancellation law holds from both sides in a set, it forms a gro

Quote:

Originally Posted by FernandoRevilla

Let $a,b\in G$ . Suppose $G=\{a_1,\ldots,a_n\}$ where $a_1,\ldots,a_n$ are distinct. Use the left cancellation law to prove that $G=\{aa_1,\ldots,aa_n\}$ .

Quote:

Originally Posted by Nikita2011

Would you explain how you obtained this set?

Well, the elements of this set are explicitly enumerated. We choose some $a\in G$ , multiply it on the right by every element of $G$ and collect the results in a new set $G'$ . The important claim about $G'$ is that $G'\subseteq G$ because $G$ is closed under *. Further, the function $f:G\to G'$ ; $f(a_i)=aa_i$ is an injection (why?), so $G=G'$ and $f$ is a bijection. Since $f$ is in particular a surjection, for every $b\in G$ there exists an $x\in G$ such that $ax=b$ .

Quote:

Originally Posted by Nikita2011

This is how I understood your post: since for any a,b in G $ax=b$ is solvable, then by definition there exists an x that satisfies that relation, so if let a=b, that proves the existence of $e_R$

Yes, but not immediately. You have a solution $e_R$ to the equation $ae_R=a$ for some particular $a$ . However, you need to show $be_R=b$ for all $b\in G$ .