Another way to describe this division is to let " e " be a index 3 of the multiplicative group ( \ Z / p \ Z ) ^ { \ times } and the other two are its cosets.
42.
The cosets of " N " are the sets of matrices with a given determinant, and hence " G " / " N " is isomorphic to the multiplicative group of non-zero real numbers.
43.
If the algebraic group is the multiplicative group of a quadratic extension of " N ", the result is the " p " + 1 method; the calculation involves pairs of numbers modulo " N ".
44.
10 ) Discussion of the multiplicative groups of ( finite ) prime fields should make it clearer up front that ( i ) " non-zero " integers modulo a prime are considered with ( ii ) multiplication as the operation.
45.
Suppose that p " 1, where p is prime, is a power of 2 and let g be a primitive root mod p, i . e . a generator for the multiplicative group of non-zero residues mod p.
46.
If we represent } } as the multiplicative group of matrices } }, where is a reflection of-dimensional space that keeps the origin fixed ( i . e ., an orthogonal matrix with determinant representing an involution ), then ?
47.
Where "'G "'m denotes the multiplicative group, plays a significant role in Hodge theory, since the Tannakian category of real Hodge structures is equivalent to the category of representations of "'S " '.
48.
Being direct product :, where } is the real multiplicative group, while if is even, these subgroups intersect in, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar :.
49.
This statement says nothing for the case of finite fields, for which there is a computational theory dedicated to finding a generator of the multiplicative group of the field ( a cyclic group ), which is " a fortiori " a primitive element.
50.
Zeta functions of Shimura varieties associated with the group " GL " 2 over other number fields and its inner forms ( i . e . multiplicative groups of quaternion algebras ) were studied by Eichler, Shimura, Kuga, Sato, and Ihara.
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