2E6 (mathematics)

In mathematics, ²E₆ is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E₆, depending on a quadratic extension of fields K⊂L. Unfortunately the notation for the group is not standardized, as some authors write it as ²E₆(K) (thinking of ²E₆ as an algebraic group taking values in K) and some as ²E₆(L) (thinking of the group as a subgroup of E₆(L) fixed by an outer involution).

Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced independently by Tits (1958) and Steinberg (1959).

Over finite fields

The group ²E₆(q²) has order q³⁶ (q¹² − 1) (q⁹ + 1) (q⁸ − 1) (q⁶ − 1) (q⁵ + 1) (q² − 1) /(3,q + 1). This is similar to the order q³⁶ (q¹² − 1) (q⁹ − 1) (q⁸ − 1) (q⁶ − 1) (q⁵ − 1) (q² − 1) /(3,q − 1) of E₆(q).

Its Schur multiplier has order (3, q + 1) except for ²E₆(2²), when it has order 12 and is a product of cyclic groups of orders 2,2,3. One of the exceptional double covers of ²E₆(2²) is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.

Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),structure,space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.

Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

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See also

Junior Certificate

The Junior Certificate (Irish: Teastas Sóisearach) is an educational qualification awarded in Ireland by the Department of Education and Skills to students who have successfully completed the junior cycle of secondary education, and achieved a minimum standard in their Junior Certification examinations. These exams, like those for the Leaving Certificate, are supervised by the State Examinations Commission. A "recognised pupil"<ref name"">Definitions, Rules and Programme for Secondary Education, Department of Education, Ireland, 2004</ref> who commences the Junior Cycle must reach at least 12 years of age on 1 January of the school year of admission and must have completed primary education; the examination is normally taken after three years' study in a secondary school. Typically a student takes 9 to 13 subjects – including English, Irish and Mathematics – as part of the Junior Cycle. The examination does not reach the standards for college or university entrance; instead a school leaver in Ireland will typically take the Leaving Certificate Examination two or three years after completion of the Junior Certificate to reach that standard.

2E6

2E6 may refer to: