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

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

    Quote Originally Posted by Nikita2011 View Post
    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 ?
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    Re: Prove that if the cancellation law holds from both sides in a set, it forms a gro

    Quote Originally Posted by FernandoRevilla View Post
    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?

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

    Quote Originally Posted by FernandoRevilla View Post
    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 View Post
    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 View Post
    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.
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