# A Long Way from Euclid by Constance Reid By Constance Reid

This full of life advisor by way of a sought after historian makes a speciality of the function of Euclid's Elements in mathematical advancements of the final 2,000 years. No mathematical historical past past common algebra and airplane geometry is important to understand the clear and easy motives, that are augmented by means of greater than eighty drawings. 1963 version.

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Extra info for A Long Way from Euclid

Example text

Then G = H + A + B. Now A ⊆ K and the sum K + B is direct. 1 (on page 38), it follows that A + B is non-periodic, as required. We are able to extend a bad factorization to a larger group from a subgroup. 3 If a proper subgroup H of a group G is k-bad, then G is both k-bad and (k + 1)-bad. PROOF There exists a factorization H = A1 +· · ·+Ak , where each subset Ai is non-periodic. 1 (on page 38), there exists a non-periodic subset C such that G = H + C. Then G = A1 + · · · + Ak + C and so G is © 2009 by Taylor & Francis Group, LLC Non-periodic factorizations 41 (k + 1)-bad.

Then the sum (i) A1 + B is a (|B| − 1)-fold, (ii) A + B1 is a (|A| − 1)-fold, (iii) A1 + B1 is a (|A| − 1)(|B| − 1)-fold © 2009 by Taylor & Francis Group, LLC New factorizations from old ones 25 factorization of G. PROOF (i) Note that A1 + B = (G\ A)+ B = (G+ B)\ (A+ B). The sum G + B is a |B|-fold factorization of G. The sum A + B is a 1-fold factorization of G. Therefore the sum A1 + B is a (|B| − 1)-fold factorization of G. The (ii) and (iii) cases can be settled in a similar way. We extend the concept of factorization to the case of more than two factors.

Let G be a finite abelian group and let A, B be subsets of G. If each element g ∈ G can be represented in exactly k ways in the form g = a + b, a ∈ A, b ∈ B, then we say that the sum A + B is a k-fold factorization of G. In other words, for each g ∈ G, there are k distinct pairs (a1 , b1 ), . . , (ak , bk ), ai ∈ A, bi ∈ B such that g = a1 + b1 = · · · = ak + bk . Further, g = a + b, a ∈ A, implies that (a, b) ∈ {(a1 , b1 ), . . , (ak , bk )}. 14 Let f : G → H be a homomorphism from G onto H such that k = |Kerf | is finite. 