You are hired by JLA Enterprise to conduct a Digital Media Investigation after it was discovered the Chief Financial Officer was embezzling money from the company’s accounts. Your job is to review all digital media involved in the case to determine if a crime has been committed. Here are some things to consider when explaining what happened during the investigation:

• Who was involved in the embezzlement? • How much money was taken? • How can you prove the individual took the money? • What programs were used to embezzle the money? • When did they start embezzling the money? • Were they working with anyone else?

Sample Solution

Mark Anderson 1.1 group concept institution theory became derived from 3 different regions of mathematics, quantity principle, the idea of algebraic equations and geometry. the first prominent mathematicians credited with studying institution concept had been A-L.Cauchy, E.Galois and J-L.Lagrange. even though Lagrange’s paintings with agencies is probably the earliest research into companies, in the 18th century, his paintings became alternatively remoted and the mid-19th century works of Cauchy and Galois are frequently taken into consideration to be the foundation of the observe of group concept. In his 1770 paper, Lagrange became the first mathematician to examine variations. His goal for the take a look at changed into to find out why cubic and quartic equations might be solved the usage of the idea of algebra. throughout his work, even as obtrusive permutation group concept is being utilized in his work, the diversifications are by no means composed and he never discusses organizations themselves. Cauchy posted his first paper on the topic of variations in 1815, but, it become now not until his work in 1844 that permutations were considered a topic in its very own right via introducing most of the key aspects of permutation companies such as the notation of high quality and poor powers of groups, identifying the electricity 0 being the identification, the cycle and permutation notation of a collection and the order of a permutation. He also proved the conjugacy of diversifications if the variations have the identical cycle structure and Cauchy’s theorem “If a high divides the order of a collection, that group has a subgroup of order “. Galois’ had papers published posthumously in 1846 with the aid of Liouville after Liouville saw a connection between Galois paintings and the permutation paintings of Cauchy from 1844. This paintings confirmed that Galois understood the relationship among the structure of a collection of diversifications related to the equation and the algebraic answer of an equation. To absolutely display this he created the belief of a normal subgroup. This become the first time the time period “group” have been used in a technical sense. 1.2 Wallpaper corporations The origins of the look at of wallpaper businesses started out because the take a look at of crystallography which was to decide the shape of crystalline solids on the atomic stage. It became from this examine that among the proofs for symmetry had been observed and set the principles for extra advanced symmetry businesses such as space institution which led to the proof of wallpaper companies. In 1830, J.F.C Hessel observed the maximum unique mixtures of reflections and rotations of a crystal round a set factor such that the photograph created is symmetrical to be 32. He proves this the usage of the regulation of indices which states “that the intercepts, OP, OQ, OR, of the natural faces of a crystal shape with the unit-cell axes a, b, c are inversely proportional to top integers, h, ok, l.” (IUCR, 2016). this is illustrated inside the photo beneath. All 32 combos with crystallographic symmetry have been then located geometrically in 1835 with the aid of M.L Frankenheim. the use of the principle of crystal instructions determined by using Hessel, A. Bravais systemized the concept and labeled the 14 spacial lattices, which we now recognize as Bravais Lattices. these lattices are defined as countless arrays of discrete factors in a 3-Dimensional aircraft created by using a set of operations defined by way of using both Hessel’s and Bravais’ work E. Fedorov and A.M Schönflies proved the life of the 230 space organizations in 1891. these space businesses are Bravais lattices which have been meditated or turned around in any of the 32 specific factor corporations observed by Hessel. those area businesses inside a 2-dimensional plane are the 17 wallpaper organizations, which despite the fact that known for hundreds of years became only proved after the proof of area businesses changed into already finished. a collection is described as a non-empty set below a binary operation, i.e. addition, multiplication and so forth. as a way to be categorized as a collection, it ought to maintain four conditions: closure, associativity, identification and inverse. permit be a collection with a binary operation Closure: For every element belonging to the institution , the end result of each elements under the binary operation on the group is same to any other element of the group. i.e. then . Associativity: If 3 factors belong to the organization then the order the operation is finished on the 3 elements will no longer have an effect on the outcome. i.e. If the . identity: There exists an element within the institution such that once the binary operation is carried out to it and any other element inside the group, the final results is same to the other detail. i.e. such that . Inverse: For each detail inside the institution there may be another detail in such that when the two factors are underneath the operation the final results is equal to the identity. i.e. such that A easy instance of a collection is the organization of integers under the operation of addition (. this could be proved via showing the organization satisfies the four axioms as stated above. Any integer delivered to every other is every other integer, so the group is closed. Addition is associative, the identification of the group is zero as any , and the inverse of any integer is as . A simple symmetry institution to recognize how symmetry is associated with institution principle is the symmetries of the rectangle. This organization consists of the linear modifications that go away the rectangles origin in place i.e. rotations and reflections. This indicates there are 4 symmetries of the rectangle as proven in determine 2. https://i.gyazo.com/4f56c4d8a306eabd9366d4a3d90a2303.png this could be displayed in more than one approaches consisting of Cayley tables, matrices and as variations References global Union of Crystallography, 2016. regulation of Rational Indices. [Online] available at: http://reference.iucr.org/dictionary/Law_of_rational_indices [Accessed 26 February 2017]. Kleiner, I., 2004. The Evolution of organization concept: A brief Survey. [Online] available at: https://www.math.lsu.edu/~adkins/m7200/GroupHistory.pdf [Accessed 24 February 2017]. O’Connor, J. J. & Robertson, E. F., 1996. The improvement of institution concept. [Online] to be had at: http://www-businesses.dcs.st-and.ac.united kingdom/records/HistTopics/Development_group_theory.html [Accessed 24 February 2017].>

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