Algebraic Structures [Lecture notes] - download pdf or read online

By Thomas Markwig Keilen

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Extra info for Algebraic Structures [Lecture notes]

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Draw the equivalence classes of (1, 1) and (−2, 3) in the plane R2. 12 (The projective line) We define for v = (v1, v2), w = (w1, w2) ∈ R2 \ {(0, 0)} v∼w ⇐⇒ ∃ λ ∈ R \ {0} : v = λ · w where λ · w := (λ · w1, λ · w2). a. Show that ∼ is an equivalence relation on M = R2 \ {(0, 0)}. It is usual to denote the equivalence class (v1, v2) of (v1, v2) by (v1 : v2), and we call the set M/ ∼ of equivalence classes the projective line over R. We denote it by P1R . b. We define on P1R a binary operation by (v1 : v2) · (w1 : w2) := (v1 · w1 − v2 · w2 : v1 · w2 + v2 · w2).

16 The formula in the above exercise is particularly useful when the set U · V is indeed a subgroup of G. This is, however, not always the case as we can easily deduce from the Theorem of Lagrange: the product of the subgroups (1 2) and (1 3) of S3 is due to the exercise a subset of cardinality 4 and by Lagrange’s Theorem it thus cannot be a subgroup of S3. 29). 17 If (G, ·) is a group and |G| is a prime number then G is cyclic. 18 Find all subgroups of the group D8 = (1 2 3 4), (2 4) . 19 Find all subgroups of the group D10 = (1 2 3 4 5), (1 5) ◦ (2 4) .

Was a group, we may very well ask if there is a natural way to pass this structure on to the set of equivalence classes. Concretely, if G is a group and U is a subgroup, is there a natural way to define a group operation on G/U? The natural should mean that the definition is somehow obvious. Given two cosets gU and hU we would want to define their product. There is of course a natural way to do this; both are subsets of G and we know already how to define the product of such subsets. What we do not know yet is if this product gives a coset again.

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Algebraic Structures [Lecture notes] by Thomas Markwig Keilen

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