( which concerns the question of bounding the size of a group if there are fixed bounds both on the order of all of its elements and the number of elements needed to generate it ) and for Burnside's lemma ( a formula relating the number of orbits of a permutation group acting on a set with the number of fixed points of each of its elements ) though the latter had been discovered earlier and independently by Frobenius and Cauchy.

* Burnside's lemma

For each g in G let X < sup > g </ sup > denote the set of elements in X that are fixed by g. Burnside's lemma asserts the following formula for the number of orbits, denoted | X / G |:

Consequently, this lemma is sometimes referred to as the lemma that is not Burnside's.

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** Burnside's lemma, a theorem of group theory

* Burnside's lemma

The central part of Burnside's group theory work was in the area of group representations, where he helped to develop some of the foundational theory, complementing and sometimes competing with the work of Frobenius, who began the subject in the 1890s.

Burnside's demeanour towards his superior is insubordinate and sometimes even hostile.

Hooker embarked on a reorganization of the army, doing away with Burnside's grand division system, which Hooker considered unwieldy ; he also no longer had sufficient senior officers on hand that he could trust to command multi-corps operations.

Burnside is also remembered for the formulation of Burnside's problem

They also fought at the Battle of Antietam, where Hartranft led its famous charge across Burnside's Bridge, suffering 120 casualties.

Parke also held significant field commands during Burnside's North Carolina Expedition, Vicksburg and the battle of Fort Stedman as well as brief stints in command of the Army of the Potomac.

The Battle of South Mills, also known as the Battle of Camden, took place on April 19, 1862 in Camden County, North Carolina as part of Union Army General Ambrose E. Burnside's North Carolina expedition during the American Civil War.

While Burnside's agents were purchasing the gunboats they were also buying or leasing other vessels to be used as transports.

Waterfall Gully was also the site of Burnside's " first secondary industry ".

This cumbersome arrangement contributed to his slowness in attacking and crossing what is now called " Burnside's Bridge " on the southern flank of the Union line.

On May 7, 1864, his brigade was called forward and smashed into Maj. Gen. Ambrose Burnside's IX Corps, which was attempting to outflank Ramseur's corps.

This follows from Burnside's p-q theorem.

after Burnside's original proof ), and a theorem of Richard Brauer and Michio Suzuki stating that a finite simple group cannot have a generalized quaternion group as its Sylow 2 subgroup.

** Burnside's theorem, a theorem of group theory

In mathematics, Burnside's theorem in group theory states that if G is a finite group of order

Burnside's theorem has long been one of the best-known applications of representation theory to the theory of finite groups, though a proof avoiding the use of group characters was published by D. Goldschmidt around 1970.

Later joint work of Ol ' shanskii and Ivanov established a negative solution to an analogue of Burnside's problem for hyperbolic groups, provided the exponent is sufficiently large.

Thus if G is any finitely generated group of exponent n, then G a homomorphic image of B ( m, n ), where m is the number of generators of G. Burnside's problem can now be restated as follows:

The full solution to Burnside's problem in this form is not known.

A famous class of counterexamples to Burnside's problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite cyclic group, the so-called Tarski Monsters.

In 1982 Ol ' shanskii was able to strengthen his results to establish existence, for any sufficiently large prime number p ( one can take p > 10 < sup > 75 </ sup >) of a finitely generated infinite group in which every nontrivial proper subgroup is a cyclic group of order p. In a paper published in 1996, Ivanov and Ol ' shanskii solved an analogue of Burnside's problem in an arbitrary hyperbolic group for sufficiently large exponents.

( This is the first edition ; the introduction to the second edition contains Burnside's famous volte face regarding the utility of representation theory.

Burnside's personal and professional life come together in Sir Geoffrey Wellingham ( Alan MacNaughtan ), who is both Burnside's former father-in-law and the Permanent Undersecretary of State at the Foreign and Commonwealth Office that oversees SIS.

Burnside's personal assistant Diane Lawler ( Elizabeth Bennett ) has regular clashes with her boss but is fiercely loyal to him.

Novikov is known for his work on combinatorial problems in group theory: the word problem for groups, and Burnside's problem.

The attack on Burnside's conjecture was started by, who studied CA groups ; these are groups such that the Centralizer of every non-trivial element is Abelian.

The creek is noted for numerous well-preserved stone arch bridges dating to the 19th century that still traverse the creek, the most famous of which is the Burnside's Bridge in the Antietam National Battlefield.

They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.

* Robinson-Schensted algorithm, giving a proof of Burnside's formula for the symmetric group.

** Burnside's problem, about the sizes of finite group mathematics