Prove that each element of G appears precisely once in each row and each column of the binary table of G?

4-6 Cancelation Properties:

Let G be a group and let x,y,z be in G.

4.6.1 Theorem.

(i) If xy=xz, then y=z

(ii) If xz=yz, then x=y.

Proof. (i) If xy=xz, then y=ey=(x^-1)xy=(x^-1)xz=ez=z

(Second Proof: xy=xz => (x^-1)xy=(x^-1)xz => ey=ez => y=z.)

(ii) Similar.

The statements (i) and (ii) are the left and right cancelation properties, respectively.

From Intro to Abstract Algebra 1! Thanks in advance! Have a blessed day! :)

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  • 6 years ago
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    PF:

    Let G be a group and let x,y,z be in G, such that

    If xy=xz, then y=z

    If xz=yz, then x=y.

    holds.

    Let us consider the column of an arbitrary element, x of G.

    Now assume there are two rows of y and z which have equal values.

    So

    xy=xz

    but by the cancellation principle that implies y=z

    so there is only one row in column x that has that value.

    the row proof is the same but switch the order of the operation around

    To prove that all the elements appear precisely once you'll use the pigeon hole principle noting that no pair of elements map to the same element under the operation so since it is closed it must hit every element in the group G.

    hope that is helpful.

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