mathematics

George Abram Miller

George Abram Miller is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand George Abram Miller rather than just read about it. In short: George Abram Miller (31 July 1863 – 10 February 1951) was an American mathematician, an early group theorist. Biography At the age of seventeen, Miller began school-teaching to raise funds for higher education.

Key takeaways

George Abram Miller belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect George Abram Miller to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of George Abram Miller from memory before moving on to harder problems.

Reference excerpt

George Abram Miller (31 July 1863 – 10 February 1951) was an American mathematician, an early group theorist.

Biography At the age of seventeen, Miller began school-teaching to raise funds for higher education. In 1882, he entered Franklin and Marshall Academy, and progressed to Muhlenberg College in 1884. He received his B.A. in 1887 and M.A. in 1890. While a graduate student, Miller was Principal of schools in Greeley, Kansas and then professor of mathematics as Eureka College in Eureka, Illinois. He corresponded with Cumberland University in Lebanon, Tennessee for his Ph.D. in 1892. He then joined Frank Nelson Cole at the University of Michigan and began to study groups. In 1895, he traveled to Europe, where he heard Sophus Lie lecture at Leipzig and Camille Jordan at Paris. In 1897, he traveled to Cornell University, where he worked as an assistant professor, and, in 1901, went to Stanford University, where he worked as associate professor. In 1906, he went to the University of Illinois, where he taught until his retirement in 1931. Miller helped in the enumeration of finite groups of degree 8, 9, and 10. Arthur Cayley had listed 198 groups of degree 8 in 1891, and Miller found two more making the total 200 in 1893. Camille Jordan had given a list for degree 9 in 1872, re-examined by Cole, and brought up to 258 groups by Miller. In 1894 Miller produced a list of 294 intransitive groups of degree 10. In consequence, the Academy of Science of Cracow awarded a prize and "Miller came to prominence in the mathematical world abruptly." Miller was president of the Mathematical Association of America from 1921 to 1922 and gave a plenary address at the International Congress of Mathematicians in 1924 in Toronto. Miller's Collected Works were edited by Henry Roy Brahana and published by University of Illinois Press, the first two volumes appearing in 1935 and 1939. The final three volumes were published in 1946, 1955, and 1959. His doctoral students include H. L. Rietz.

Publications 1892: An introduction to the study of Determinants, with examples and applications. 1905: Groups of subtraction and division. 1908: "On the multiple holomorphs of a group", Mathematische Annalen 66(1): 133-142 1911: "Abstract definitions of all the substitution groups whose degrees do not exceed seven", American Journal of Mathematics 1911: The Algebraic Equation 1916: (with H. F. Blichfeldt, & L. E. Dickson) Theory and Application of Finite Groups from University of Michigan Historical Math Collection, original publisher: John Wiley & Sons. 1916: Historical Introduction to Mathematical Literature from Cornell University Historical Math Monographs, original publisher Macmillan Publishers. 1947: "An Eleventh Lesson in the History of Mathematics", Mathematics Magazine 21(1): 48-55.

References

External links

George A. Miller Papers 1895–1947, 1951 from University of Illinois Archives. O'Connor, John J.; Robertson, Edmund F., "George Abram Miller", MacTutor History of Mathematics Archive, University of St Andrews George Abram Miller at the Mathematics Genealogy Project

Illustrations

Worked examples

Example 1 — a first encounter with George Abram Miller

Start with the simplest possible case. Write down what George Abram Miller claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to George Abram Miller before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about George Abram Miller ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of George Abram Miller

In research: George Abram Miller appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses George Abram Miller in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: George Abram Miller is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1863 births, 1951 deaths, American historians of mathematics, so understanding it makes those chapters shorter.
In everyday life: Look for George Abram Miller outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study George Abram Miller in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what George Abram Miller means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
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Frequently asked questions

What is George Abram Miller in simple terms?

George Abram Miller (31 July 1863 – 10 February 1951) was an American mathematician, an early group theorist. Biography At the age of seventeen, Miller began school-teaching to raise funds for higher education.

Why does George Abram Miller matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study George Abram Miller?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on George Abram Miller.