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The principal mathematical achievements of the Renaissance and the following period were in algebra (see Chapter 1), and even medieval Arab (or, to be more exact, Arabic-language) mathematics was oriented towards algebra rather than geometry (see Chapter 1 and Note 9). The seventeenth century was marked by the development of the calculus, which for centuries afterwards was regarded as the principal branch of mathe­ matics. , differential equations). At the present time this term sounds absurd (surely, probability theory or mathematical logic cannot now be regarded as branches of mathematics "lower" than the calculus) but is nevertheless widely used, especially in Russian­ and German-speaking countries.

He worked there for more than fifty years, until his death in 1 868; he rose from the rank of a lower staff member to the position of observatory director (he combined the latter job at the end of his life with professorial duties at Leipzig University). Mobius's whole life passed within the walls of the Pleisenburg Observatory; his study, the flat in which he lived with his wife and children, and the hall where he was always glad to lecture, were all in this building. It was characteristic of Mobius that he took his observatory duties very seriously.

1 fJ e Jt' and therefore p-1 rx. Jt' and fJJt' have no common elements. Jt', fJJt', y:K, . . , fJ, y, . . are all the elements of t:§. Some of these sets coin­ cide, and some have no common elements. Jt', fJJt', . . etc. (including the subgroup Jt' itself which may be written in the form e:K or in the form y:K for any element y in Jt'). Such a partition of t:§ into nonoverlapping subsets (see the schematic Fig. 3) is known as a partition of t:§ into left cosets of the group t:§ with respect to the subgroup Jt'.

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A Characterization of Alternating Groups by Mazurov V.D.

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