Explore fundamental concepts of Group Theory: Identity, Inverses, Associativity, and Closure.
Abstract Settings
Axiom Verification
Closure
Associativity
Identity
Inverse
Group Properties
Cyclic Group (Z₄)
Selected Structure
Definition and structural properties of the selected group.
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Order of Group4
Is Abelian?Yes
Identity Element (e)0
OperationAddition modulo 4
Elements of G
0123
About this calculator
What is Group Theory?
Group Theory is the study of symmetry and abstract algebraic structures called 'groups'. A group is a set G combined with an operation * that satisfies four axioms: Closure, Associativity, Identity element, and Inverse element.
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Pro Tips
A group is 'Abelian' if the operation is commutative: a*b = b*a.
The 'Order' of a group is the number of elements it contains.
Subgroups are subsets of groups that are groups themselves under the same operation.
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Fun Facts
"Group theory is used to solve the Rubik's Cube efficiently."
"In particle physics, the Standard Model is based on symmetry groups like SU(3) × SU(2) × U(1)."
"Évariste Galois developed group theory in the early 1800s to determine whether polynomial equations could be solved by radicals."
Formula: Group Axions: ∀a,b ∈ G: (a*b) ∈ G, (a*b)*c = a*(b*c), e*a = a, a*a⁻¹ = e
