Cyclic group and Point groups in three dimensions

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Difference between Cyclic group and Point groups in three dimensions

Cyclic group vs. Point groups in three dimensions

In algebra, a cyclic group or monogenous group is a group that is generated by a single element. In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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Cycle graph (algebra)

In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.

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Dicyclic group

In group theory, a dicyclic group (notation Dicn or Q4n) is a member of a class of non-abelian groups of order 4n (n > 1).

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Direct product of groups

In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.

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Frieze group

In mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction.

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Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).

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Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.

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Orbifold notation

In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.

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Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

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Rotational symmetry

Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.

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Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

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Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.

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