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In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.[1]
^Barnard, Tony & Neil, Hugh (2017). Discovering Group Theory: A Transition to Advanced Mathematics. Boca Ratan: CRC Press. p. 94. ISBN 9781138030169.
the definition of an isomorphism is quite natural. An isomorphism of groups may equivalently be defined as an invertible group homomorphism (the inverse...
mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship...
isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique. The term isomorphism is mainly used for algebraic...
Babai, László (January 9, 2017), Graph isomorphism update Sun, Xiaorui (2023). "Faster Isomorphism for p-Groups of Class 2 and Exponent p". arXiv:2303...
science: Can the graph isomorphism problem be solved in polynomial time? (more unsolved problems in computer science) The graph isomorphism problem is the computational...
mathematics contains the finite groups of small order up to groupisomorphism. For n = 1, 2, … the number of nonisomorphic groups of order n is 1, 1, 1, 2,...
of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets...
abelian groups, then the set of isomorphism classes of extensions of Q {\displaystyle Q} by a given (abelian) group N {\displaystyle N} is in fact a group, which...
an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space. There is a natural group homomorphism p from E(n)...
topological groups is continuous if and only if it is continuous at some point. An isomorphism of topological groups is a groupisomorphism that is also...
Isomorphism problem may refer to: graph isomorphism problem groupisomorphism problem This disambiguation page lists mathematics articles associated with...
in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs...
{\displaystyle \eta _{X}} is an isomorphism in D {\displaystyle D} , then η {\displaystyle \eta } is said to be a natural isomorphism (or sometimes natural equivalence...
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping...
permutation groups) if there exists a bijective map λ : X → Y and a groupisomorphism ψ : G → H such that λ(f1(g, x)) = f2(ψ(g), λ(x)) for all g in G and...
logarithm is a group isomorphism of this group to the additive group of real numbers, R {\displaystyle \mathbb {R} } . The multiplicative group of a field F {\displaystyle...
subgroups of G this induces an isomorphism from G to a subgroup of the permutation group of degree [G : H]. In every group G, conjugation is an action of...
The isomorphism L2(9) ≅ A6 allows one to see the exotic outer automorphism of A6 in terms of field automorphism and matrix operations. The isomorphism L4(2)...
is the groupisomorphism problem, which asks whether two groups given by different presentations are actually isomorphic. For example, the group with presentation...
corresponding curves are isomorphic. Isomorphism classes of elliptic curves are specified by the j-invariant. The isomorphism classes can be understood in a...
N\rtimes _{\varphi }H=H\ltimes _{\varphi }N} denotes a group G, unique up to a groupisomorphism, which is a semidirect product of N and H, with the commutation...
{1}}+b\,{\hat {i}}+c\,{\hat {j}}+d\,{\hat {k}}} This map is in fact a groupisomorphism. Additionally, the determinant of the matrix is the squared norm of...
amounts to a groupisomorphism log b : H → Z n , {\displaystyle \log _{b}\colon H\to \mathbf {Z} _{n},} where Zn denotes the additive group of integers...
The isomorphism can be described as follows: given a group representation ρ : G → GL(V), where V is a vector space over k, GL(V) is the group of its...
typical realization of this group is as the complex nth roots of unity. Sending a to a primitive root of unity gives an isomorphism between the two. This can...
of logarithms is the story of a correspondence (in modern terms, a groupisomorphism) between multiplication on the positive real numbers and addition...
Logarithmic functions are the only continuous isomorphisms between these groups. By means of that isomorphism, the Haar measure (Lebesgue measure) dx on...
correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups. A Lie group G {\displaystyle...
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