Definition Of Group Work Reflection

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Reflection of Group Project

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Compare Accounts. The offers that appear in this table are from partnerships from which Investopedia receives compensation. This compensation may impact how and where listings appear. Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted. Individual axioms may be "weakened" to assert only the existence of a left identity and left inverses.

From these one-sided axioms , one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are no weaker. Therefore it is customary to speak of the identity. Therefore it is customary to speak of the inverse of an element. When studying sets, one uses concepts such as subset , function, and quotient by an equivalence relation. When studying groups, one uses instead subgroups , homomorphisms , and quotient groups. These are the appropriate analogues that take into account the existence of the group structure.

Group homomorphisms [d] are functions that respect group structure; they may be used to relate two groups. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation. An isomorphism is a homomorphism that has an inverse homomorphism; equivalently, it is a bijective homomorphism.

The collection of all groups, together with the homomorphisms between them, form a category , the category of groups. Knowing a group's subgroups is important in understanding the group as a whole. Again, this is a subgroup, because combining any two of these four elements or their inverses which are, in this particular case, these same elements yields an element of this subgroup. In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed.

In some situations the set of cosets of a subgroup can be endowed with a group law, giving a quotient group or factor group. For this to be possible, the subgroup has to be normal. The group operation on the quotient is shown in the table. Quotient groups and subgroups together form a way of describing every group by its presentation : any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations.

In general, homomorphisms are neither injective nor surjective. The kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon. Examples and applications of groups abound. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra. Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups.

The second image shows some loops in a plane minus a point. The blue loop is considered null-homotopic and thus irrelevant , because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop or any other loop winding once around the hole. This way, the fundamental group detects the hole. In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background.

In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory , in particular when implemented for finite groups. Many number systems, such as the integers and the rationals enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields.

Further abstract algebraic concepts such as modules , vector spaces and algebras also form groups. The desire for the existence of multiplicative inverses suggests considering fractions a b. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. The rational numbers including zero also form a group under addition. Group theoretic arguments therefore underlie parts of the theory of those entities. A familiar example is addition of hours on the face of a clock , where 12 rather than 0 is chosen as the representative of the identity. Hence all group axioms are fulfilled. The group operation is multiplication of complex numbers. Some cyclic groups have an infinite number of elements.

The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups ; and reflecting this state of affairs, many group-related notions, such as center and commutator , describe the extent to which a given group is not abelian. Symmetry groups are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory see below.

A group is said to act on another mathematical object X if every group element can be associated to some operation on X and the composition of these operations follows the group law. For example, an element of the 2,3,7 triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles. In chemical fields, such as crystallography , space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that molecules sometimes can change symmetry.

The Jahn—Teller effect is a distortion of a molecule of high symmetry when it adopts a particular ground state of lower symmetry from a set of possible ground states that are related to each other by the symmetry operations of the molecule. Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition , for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons. Finite symmetry groups such as the Mathieu groups are used in coding theory , which is in turn applied in error correction of transmitted data, and in CD players. Matrix groups consist of matrices together with matrix multiplication.

The dihedral group example mentioned above can be viewed as a very small matrix group. Rotation matrices in this group are used in computer graphics. Representation theory is both an application of the group concept and important for a deeper understanding of groups. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.

A group action gives further means to study the object being acted on. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups , especially locally compact groups. Galois groups were developed to help solve polynomial equations by capturing their symmetry features. Analogous Galois groups act on the solutions of higher-degree polynomials and are closely related to the existence of formulas for their solution. Abstract properties of these groups in particular their solvability give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and roots similar to the formula above. Modern Galois theory generalizes the above type of Galois groups by shifting to field theory and considering field extensions formed as the splitting field of a polynomial.

This theory establishes—via the fundamental theorem of Galois theory —a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics. A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. The order of an element equals the order of the cyclic subgroup generated by this element. The Sylow theorems give a partial converse. The first part will be drawing upon through the appropriate literature, and there are four points to prove the importance of critical reflection in personal development.

It will be expounded and distinguish Reflection in action and Reflection on action. Then, the theory of Single and double loop learning will be introduced. Moreover, the reflection also includes the difficulties and challenges. Word count: 2. In the context of human reflection, the definition can be extended to cognitive acts such as observation, thinking, consideration, memorizing, contemplation and meditation. Reflective practice, in its primary variation, could be explained as thinking about or reflecting on what one do.

It is closely related to the concept of drawing lessons from experience, in that one. Once receiving notification I was both pleasantly surprised and excited. This is in-part due to my previous credentials which include professional experience in leadership roles in the hospitality industry, beauty industry and welfare-to-work sector. Based on my working history, the course is everything I expected it to be as I have always had a keen interest for gaining a better understanding of myself and others.

My only criticism, is that I personally feel I would have gained more from. Collaborative learning involves small groups of students who act on a structured learning activity to solve a problem, complete a task, or create a product. In simpler terms, the students work in groups to learn or understand a new concept they are studying. In this learning approach, it is important to understand that the students are accountable for individual work as well as the group work they do as a group in collaborative learning MY VISION My vision of collaborative. According to Literary Devices, , writing can also be described as a voice that readers listen to when they read the work of a writer.