Mathematical structure with multiplication as its operation
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In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
the group under multiplication of the invertible elements of a field,[1] ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field F, the group is (F ∖ {0}, •), where 0 refers to the zero element of F and the binary operation • is the field multiplication,
the algebraic torus GL(1).[clarification needed].
^See Hazewinkel et al. (2004), p. 2.
and 22 Related for: Multiplicative group information
mathematics and group theory, the term multiplicativegroup refers to one of the following concepts: the group under multiplication of the invertible...
integers form a group under multiplication modulo n, called the multiplicativegroup of integers modulo n. Equivalently, the elements of this group can be thought...
solution, i.e., when it exists, a modular multiplicative inverse is unique: If b and b' are both modular multiplicative inverses of a respect to the modulus...
from an orthogonal group over a field F to the quotient group F× / (F×)2 (the multiplicativegroup of the field F up to multiplication by square elements)...
\mathrm {G} _{m}} is called multiplicativegroup, because its k {\displaystyle k} -points are isomorphic to the multiplicativegroup of the field k {\displaystyle...
generalizations See Multiplication in group theory, above, and multiplicativegroup, which for example includes matrix multiplication. A very general, and...
1{\pmod {n}}} . In other words, the multiplicative order of a modulo n is the order of a in the multiplicativegroup of the units in the ring of the integers...
spectrum of a group ring. More generally, one can form groups of multiplicative type by letting A be a non-constant sheaf of abelian groups on S. For a...
power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group. Every infinite...
mathematics, the circle group, denoted by T {\displaystyle \mathbb {T} } or S 1 {\displaystyle \mathbb {S} ^{1}} , is the multiplicativegroup of all complex numbers...
mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity...
{\begin{pmatrix}a&b\\0&1\end{pmatrix}}\mapsto a^{u}} is a group homomorphism. Consider a multiplicativegroup of positive real numbers (R+, ⋅) for any complex...
rings and fields to distinguish the additive underlying group from the multiplicativegroup of the invertible elements. In older terminology, an additive...
Mathieu group on a 24-set and in studying symmetry in certain models of finite geometries. The quaternions with norm 1 (the versors), as a multiplicative group...
conventions for abelian groups – additive and multiplicative. Generally, the multiplicative notation is the usual notation for groups, while the additive...
logarithmic derivative lie two basic facts about GL1, that is, the multiplicativegroup of real numbers or other field. The differential operator X d d X...
operation instead of multiplicative notation). For a general subgroup H {\displaystyle H} , it is desirable to define a compatible group operation on the...
linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because...
a primitive root modulo n if and only if g is a generator of the multiplicativegroup of integers modulo n. Gauss defined primitive roots in Article 57...
algebraic groups, the multiplicative and additive groups, behave very differently in terms of their linear representations (as algebraic groups). Every...
3. Compare the values 6 and 4 for Euler's totient function, the multiplicativegroup of integers modulo n for n = 9 and 10, respectively. This triples...
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