Helen Popova Alderson (1924–1972) was a Soviet and British mathematician and mathematics translator known for her research on

quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ...

s and on higher

reciprocity law In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an ir ...

Life

Alderson was born on 14 May 1924 in

Baku Baku (, ; ) is the Capital city, capital and List of cities in Azerbaijan, largest city of Azerbaijan, as well as the largest city on the Caspian Sea and in the Caucasus region. Baku is below sea level, which makes it the List of capital ci ...

, then part of the

Soviet Union The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...

, to a family of two academics from

Moscow Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents with ...

. Her father, a neurophysiologist, had been a student of

Ivan Pavlov Ivan Petrovich Pavlov (, ; 27 February 1936) was a Russian and Soviet experimental neurologist and physiologist known for his discovery of classical conditioning through his experiments with dogs. Pavlov also conducted significant research on ...

. She began studying mathematics at

Moscow University Moscow State University (MSU), officially M. V. Lomonosov Moscow State University,. is a public research university in Moscow, Russia. The university includes 15 research institutes, 43 faculties, more than 300 departments, and six branches. Al ...

in 1937, when she was only 13. She had to break off her studies because of

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

, moving to Paris as a refugee with her family. After the war, she returned to study at the

University of Edinburgh The University of Edinburgh (, ; abbreviated as ''Edin.'' in Post-nominal letters, post-nominals) is a Public university, public research university based in Edinburgh, Scotland. Founded by the City of Edinburgh Council, town council under th ...

. She completed a Ph.D. there in 1951; her dissertation was ''Logarithmetics of Non-Associative Algebras''. After leaving mathematical research to raise two children in Cambridge, she was funded by the

Calouste Gulbenkian Foundation The Calouste Gulbenkian Foundation (), commonly referred to simply as the Gulbenkian Foundation, is a Portuguese institution dedicated to the promotion of the arts, philanthropy, science, and education. One of the wealthiest charitable founda ...

with a Fellowship at

Lucy Cavendish College, Cambridge Lucy Cavendish College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge. The college was established in 1965 as a women's college and since 2021 has admitted both women and men. The college is na ...

, beginning in the late 1960s. At Cambridge, she worked with

J. W. S. Cassels John William Scott "Ian" Cassels, Fellow of the Royal Society, FRS (11 July 1922 – 27 July 2015) was a British mathematician. Biography Cassels was educated at Neville's Cross Council School in Durham, England, Durham and George Heriot's ...

. She died on 5 November 1972, from complications of kidney disease.

Research

In the theory of higher

s, Alderson published necessary and sufficient conditions for 2 and 3 to be

seventh power Seventh is the ordinal form of the number seven. Seventh may refer to: * Seventh Amendment to the United States Constitution * A fraction (mathematics), , equal to one of seven equal parts Film and television *"The Seventh", a second-season epi ...

s, in

modular arithmetic In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...

modulo a given

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

p

. According to , "plain quasigroups were first studied by Helen Popova-Alderson, in a series of papers dating back to the early fifties". Smith cites in particular a posthumous paper and its references. In this context, a

is a mathematical structure consisting of a set of elements and a

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...

that does not necessarily obey the

associative law In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...

, but where (like a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

) this operation can be inverted. Being plain involves having only a finite number of elements and no non-trivial subalgebras.

Translation

As well as Russian, English, and French, Alderson spoke Polish, Czech, and some German. She became the English translator of ''Elementary Number Theory'', a textbook originally published in Russian in 1937 by B. A. Venkov. Her translation was published by Wolters-Noordhoff of Groningen in 1970. As well as the original text, it includes footnotes by Alderson updating the material with new developments in number theory.

Selected publications

References

{{DEFAULTSORT:Alderson, Helen Popova 1924 births 1972 deaths Scientists from Baku Soviet mathematicians Soviet women mathematicians British mathematicians British women mathematicians Technical translators Russian–English translators Alumni of the University of Edinburgh College of Science and Engineering Fellows of Lucy Cavendish College, Cambridge 20th-century translators Soviet emigrants to the United Kingdom Soviet expatriates in France