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Concept# Quotient

Summary

In arithmetic, a quotient (from quotiens 'how many times', pronounced ˈkwoʊʃənt) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of a general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense, and 6\tfrac{2}{3}=6.66... (a repeating decimal) in the second sense.
Ratios can be defined as dimensionless quotients;
non-dimensionless quotients are also known as rates.
Notation
Division (mathematics)#Notation
The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line. The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole.
\dfrac{1}{2} \quad
\begin{alig

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We present a contribution to the structure theory of locally compact groups. The emphasis is put on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a complete description of groups all of whose proper quotients are compact, of characteristically simple groups and of groups admitting a subnormal series with all subquotients compact, or compactly generated Abelian, or compactly generated and topologically simple. Two appendices introduce results and examples around the concept of quasi-product.

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