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Group Theory

Group theory is the study of algebraic structures known as groups. It has many important applications in physics, chemistry and material science. It also has important applications in public key cryptography. 437 more words

Group Theory

A Generalization of Wilson's Theorem (due to Gauss)

John Wilson (1741-1793) was a well-known English mathematician in his time, whose legacy lives on in his eponymous result, Wilson’s Theorem. To recall, this is the statement that an integer is prime if and only if… 1,015 more words

Number Theory

Permutation Groups vis-à-vis Conformal Maps of the Riemann Sphere

In this post, we discuss a few ways in which the symmetric and alternating groups can be realized as finite collections of self-maps on the Riemann sphere.   1,820 more words

Group Theory

Formal Groups and Where to Find Them

In 1946, S. Bochner published the paper Formal Lie Groups, in which he noted that several classical theorems (due to Sophus Lie) concerning infinitesimal transformations… 1,823 more words

Group Theory

The Orders of Simple Groups

In 1832, Galois introduced the concept of normal subgroups, and proved that the groups (for ) and (for ) were simple, i.e. admit no non-trivial (proper) normal subgroups.   1,364 more words

Number Theory

Permutations of Maximal Order

Let denote the permutation group on n letters.  Herein, we consider the following question, brought to light by Landau in 1902:

Question: What is the maximal order of an element in ? 892 more words

Number Theory

107: The Order of the Product of 2 Group Elements

Where is a group, and , it is always the case that

That is, we can speak of ‘the’ order of the product of 2 group elements without worrying about which product it is.