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The need for closure

Published online by Cambridge University Press:  06 June 2019

Christopher D. Hollings
Christopher D. Hollings*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG The Queen’s College, Oxford OX1 4AW e-mail: christopher.hollings@maths.ox.ac.uk
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When defining a group, do we need to include closure? This is a detail that is often touched upon when the notion of a group is introduced to undergraduates. Should closure be listed as an axiom in its own right, or should it be regarded as an inherent property of the binary operation? There is no clear answer to this question, although there are firm opinions on both sides. Indeed, a very brief survey of group theory textbooks found in [1, pp. 458-459] suggests that there is a rough 50 : 50 split between authors who include closure explicitly and those who do not. In this Article, we go back to the beginning of the twentieth century to provide some historical perspective on this problem.

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Articles
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The Mathematical Gazette , Volume 103 , Issue 557 , July 2019 , pp. 248 - 256
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© Mathematical Association 2019 

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