A course in abstract algebra by Nicholas Jackson

By Nicholas Jackson

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As √ before, we have the identity matrix I and the two rotation matrices A y = 3x D and B, but we also have three further matrices. A bit of examination or experimentation shows that these matrices represent reflections in lines passing through the origin: C is reflection in the x–axis, D is √ √ y=0 C reflection in the line y = − 3x, and E is reflection in the line y = 3x. This matrix group tells us how six specific geometric transformations interact with each other. 5: Reflections represented by same as an anticlockwise rotation through an angle of 3 .

How many of these determine semigroups, that is, how many are associative? How many of those determine monoids? How many determine a group structure? 3 Which of the following binary operations (defined on the set R of real numbers) are associative, and which are commutative? 4 Show that the multiplication table for a finite group G satisfies the Latin square property. That is, show that each element of the group occurs exactly once in each row and column of the table. 5 Let ∗ : R×R → R by x ∗ y = xy + 1.

The symmetric group S4 is isomorphic to the symmetry group of the tetrahedron. Later on we will meet Cayley’s Theorem, which states that any group can be regarded as a group of permutations (although not, in general, the full symmetric group Sn for some value of n). To investigate these permutation groups, we need a coherent and consistent notation, at least for permutations on finite sets. One method, given that a permutation σ : X → X is determined completely by its action on the elements of the set X, is to represent it in the form of an array: x1 σ ( x1 ) x2 σ ( x2 ) ...

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